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Engineering  Science  Series 


ALTERNATING  CURRENTS  AND  ALTERNATING 
CURRENT  MACHINERY 


ENGINEERING  SCIENCE  SERIES 


EDITED  BY 

DUGALD  C.  JACKSON,  C.E. 

Professor  of  Electrical  Engineering 
Massachusetts  Institute  of  Technology 
Fellow  and  Past  President  A.I.E.E. 


EARLE  R.  HEDRICK,  Ph.D. 

PK0FES80B  OF  MATHEMATICS,  UNIVERSITY  OF  M I8SOUP.I 
Member  A.S.M.E. 


ALTERNATING  CURRENTS  AND 
ALTERNATING  CURRENT 
MACHINERY 

NEW  EDITION,  REWRITTEN  AND  ENLARGED 


BY 

UUGALD  C.  JACKSON,  C.E. 

PROFESSOR  OF  ELECTRICAL  ENGINEERING,  MASSACHUSETTS  INSTITUTE  OF  TECHNOLOGY 
PAST-PRESIDENT  AND  FELLOW  OF  THE  AMERICAN  INSTITUTE  OF  ELECTRICAL 
ENGINEERS,  PAST-PRESIDENT  OF  THE  SOCIETY  FOR  THE  PRO- 
MOTION OF  ENGINEERING  EDUCATION,  ETC. 

AND 

JOHN  PRICE  JACKSON,  M.E.,  Sc.D. 

COMMISSIONER  OF  LABOR  AND  INDUSTRY,  COMMONWEALTH  OF  PENNSYLVANIA,  DEAN 
OF  THE  SCHOOL  OF  ENGINEERING,  PENNSYLVANIA  STATE  COLLEGE,  FELLOW 
OF  THE  AMERICAN  INSTITUTE  OF  ELECTRICAL  ENGINEERS,  MEMBER  OF 
THE  AMERICAN  SOCIETY  OF  MECHANICAL  ENGINEERS,  ETC. 


Neto  Iforfe 

THE  MACMILLAN  COMPANY 
1922 


All  rights  reserved 


Copyright,  1896,  1913, 

By  THE  MACMILLAN  COMPANY. 

Set  up  and  electrotyped.  New  edition.  Published  September,  191 
Reprinted  July,  1914;  October,  1917. 


NortoootJ  J9rfss 

J.  S.  Cushing  Co.  — Berwick  & Smith  Co. 
Norwood,  Mass.,  U.S.A. 


2-A  3/3  3 


; 3 (X^ 


PREFACE 


This  is  a new  edition  of  the  authors’  book  on  Alternating 
Currents  and  Alternating  Current  Machinery  which  was  first 
published  in  1896,  and  which  has  now  been  rewritten  and 
greatly  extended.  The  former  editions  met  so  favorable  a re- 
sponse that  the  authors  cannot  help  but  believe  that  it  served 
an  important  place  in  connection  with  the  development  of  the 
methods  of  teaching  the  subject  of  alternating  currents  and 
their  applications ; and  they  earnestly  hope  that  the  new  edi- 
tion will  be  equally  useful. 

In  this  edition  are  maintained  the  well-known  features  of 
the  earlier  book  in  which  were  worked  out  the  characteristics 
of  electric  circuits,  their  self-induction,  electrostatic  capacity, 
reactance  and  impedance,  and  the  solutions  of  alternating  cur- 
rent flow  in  electric  circuits  in  series  and  parallel.  More  atten- 
tion is  paid  to  the  transient  state  in  electric  circuits  than  was 
the  case  in  the  original  edition.  A considerable  amount  of 
related  matter  has  been  introduced  in  respect  to  vectors,  com- 
plex quantities,  and  Fourier’s  series  which  the  authors  believe 
will  be  useful  to  students  and  engineers.  The  treatment  of 
power  and  power  factor  has  been  given  great  attention,  and  a 
full  chapter  is  now  allotted  to  the  hysteresis  and  eddy  current 
losses  which  are  developed  in  the  iron  cores  of  electrical  ma- 
chinery. More  space  and  more  complete  treatment  have  been 
assigned  to  synchronous  machines  and  to  asynchronous  motors 
and  generators.  The  treatment  of  the  self-inductance  and  mutual 
inductance  of  line  circuits  and  skin  effect  in  electric  conductors 
which  was  found  in  the  old  book  has  been  extended,  and  it  has 
been  supplemented  by  a treatment  of  the  electrostatic  capacity 
of  lines  and  the  influences  of  distributed  resistance,  inductance, 
and  capacity. 

In  all  these  features  as  well  as  in  others  the  book  has  been 


Y1 


PREFACE 


brought  up  to  the  requirements  of  present  day  teaching.  The 
book  covers  the  ground  that  is  needed  to  give  a fairly  complete 
course  in  the  essential  elements  of  alternating  currents  and 
their  applications  to  machinery.  It  is  longer  than  will  be 
needed  in  the  courses  in  many  of  the  engineering  schools,  but 
chapters  may  be  selected  so  as  to  meet  the  requirements  of 
each  school.  Thus,  if  an  abbreviated  course  is  required,  Chap- 
ters V and  XIII  may  be  omitted,  and  if  the  course  does  not  go 
into  electrical  machinery,  the  first  eight  chapters  only  need  be 
considered.  In  those  colleges  where  the  students  have  a course 
in  alternating  current  measurements  previous  to  their  entering 
upon  the  subject  of  alternating  current  machinery  the  second 
chapter  may  be  omitted  during  the  study  of  this  text,  but  the 
chapter  will  be  useful  in  connection  with  the  course  in  alter- 
nating current  measurements.  The  book  is  also  serviceable  in 
connection  with  instruction  in  the  electrical  transmission  of 
power.  For  this  purpose  the  authors  suggest  more  particularly 
the  use  of  the  portions  of  Chapter  I which  deal  with  complex 
quantities  and  Fourier’s  series  and  the  whole  of  Chapters  IV, 
V,  VI,  VII,  VIII,  XIII.  These  are  all  particularly  related  to 
matters  which  are  fundamental  not  only  to  electrical  machinery 
but  also  to  the  transmission  of  power  and  the  other  branches 
in  which  alternating  currents  are  usefully  applied. 

The  authors  have  endeavored  to  make  the  phraseology  simple 
and  to  illustrate  the  applications  of  the  principles  by  examples 
drawn  from  the  best  practice  in  the  art.  As  in  the  first  edi- 
tion, original  methods  have  been  introduced  in  various  instances 
to  gain  simple  paths  to  results,  every  effort  being  made  to  pre- 
sent a full  physical  conception  of  phenomena  to  the  reader's 
mind.  The  mathematics  used  are  merely  logical  means  for 
accomplishing  the  end,  and  are  by  no  means  to  be  considered 
from  any  other  standpoint.  It  has  been  sought  by  the  authors 
to  avoid  either  the  error  of  presenting  unnecessary  formulas  or, 
on  the  other  hand,  of  giving  results  without  supporting  them 
on  reasons,  both  of  which  are  fatal  to  a student’s  effective  prog- 
ress, since  they  leave  him  without  a true  physical  conception  of 
the  phenomena  studied.  It  has  been  the  authors’  aim  to  pro- 
duce a text  which  avoids  the  fault  of  being  cursorily  descriptive 
and  at  the  same  time  avoids  the  reasonable  objections  to  a 
treatise  which  is  unrelieved  mathematics. 


PREFACE 


vii 

The  original  edition  of  the  book  was  noticeable  for  utilizing 
the  word  “ active  ” for  representing  the  true  power  component 
of  voltage  or  current,  and  that  practice,  which  has  now  been 
approved  by  the  Standards  Committee  of  the  American  Insti- 
tute of  Electrical  Engineers,  is  followed  in  this  edition,  while 
the  component  at  right  angles  is  in  this  edition  called  either 
the  quadrature  component  or  the  reactive  component,  the  latter 
phraseology  being  also  now  in  accordance  with  the  recommen- 
dations of  the  Standards  Committee  of  the  American  Institute 
of  Electrical  Engineers.  In  various  respects  the  book  contains 
original  treatments  which  will  be  obvious,  but  the  authors  have 
made  an  effort  to  set  forth  the  best  treatment  of  each  of  the 
problems  of  alternating  currents  and  therefore  have  brought 
under  contribution  most  of  the  literature  and  practice  in  the 
art  in  the  course  of  their  selections  of  methods  of  presentation. 

The  examples  which  are  introduced  at  intervals  throughout 
the  text  for  the  purpose  of  illustrating  the  way  in  which  the 
principles  may  be  utilized  will  be  useful  to  teachers  and  stu- 
dents. Footnotes  referring  mostly  to  correlated  articles  in  the 
book  itself  are  numerous.  The  many  historical  footnotes  which 
were  in  the  former  editions  have  been  mostty  omitted  from  this 
edition,  since  the  early  literature  has  been  now  far  outstripped 
and  is  no  longer  an  essential  source  of  knowledge  for  the  under- 
graduate student. 

The  book  is  placed  in  the  hands  of  electrical  engineers  in  the 
belief  that  it  will  prove  valuable  as  a textbook  in  the  engineer- 
ing schools,  and  also  prove  of  service  as  a reference  book  with 
respect  to  the  principles  used  in  the  numerous  applications  of 
alternating  currents  to  practical  purposes  which  now  make  so 
large  a part  of  electrical  engineering. 

The  authors  wish  to  express  their  sincere  thanks  to  Mr.  H. 
P.  Wood,  Professor  of  Electrical  Engineering  at  the  Georgia 
School  of  Technology,  Mr.  Charles  L.  Kinsloe,  Professor  of 
Electrical  Engineering  at  the  Pennsylvania  State  College,  and 
Mr.  Charles  W.  Green,  Instructor  in  Electrical  Engineering  at 
the  Massachusetts  Institute  of  Technology,  who  have  read  the 
proof  and  made  many  suggestions.  It  is  not  to  be  expected 
that  a book  of  the  extent  and  importance  of  this,  the  manu- 
script of  which  is  written  and  the  proof  read  in  the  midst  of 
the  exacting  employments  of  men  in  important  duty  in  engi- 


PREFACE 


viii 

neering  schools,  can  be  entirely  free  from  ambiguities  and 
errors ; but  the  authors  venture  the  hope  that  any  errors  or 
ambiguities  which  have  escaped  their  vigilance  are  few.  They 
will  greatly  appreciate  having  brought  to  their  attention  any 
which  readers  may  find. 


CONTENTS 


CHAPTER  PAGE 

I.  The  Voltage  developed  by  Alternators.  The  Use  of 

Vectors  and  the  Complex  Quantity.  Fourier’s  Series  1 

II.  Elementary  Statements  concerning  Transformers  and 

Measuring  Instruments 61 

III.  Armature  and  Field  Windings  for  Alternators.  Ma- 

terials of  Construction 77 

IV.  Self-induction,  Electrostatic  Capacity,  Reactance, 

and  Impedance  .........  128 

V.  The  Use  of  Complex  Quantities  Extended  . . . 241 

VI.  Solution  of  Circuits.  Application  of  Graphical  and 

Analytical  Methods 257 

VII.  Power.  Power  Factor 312 

VIII.  Polyphase  Circuits  and  the  Measurement  of  Power 

therein 359 

IX.  Hysteresis  and  Eddy  Current  Losses  ....  400 

X.  Mutual  Induction.  Transformers 443 

XI.  Synchronous  Machines.  Alternators,  Motors,  Rotary 

Converters,  Frequency  Changers 597 

XTI.  Asynchronous  Motors  and  Generators  ....  784 
XIII.  Self-inductance,  Mutual  Inductance,  and  Electro- 
static Capacity  of  Parallel  Wires.  Skin  Effect. 
Effects  of  Distributed  Resistance,  Inductance,  and 
Capacity.  Corona 897 

INDEX 955 


IX 


ALTERNATING  CURRENTS 


CHAPTER  I 

THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS.  THE  USE  OP 
VECTORS  AND  THE  COMPLEX  QUANTITY.  FOURIER’S 
SERIES 

1.  Direct  and  Alternating  Currents  obey  the  Same  Laws.  — A 

deeply  rooted  belief  seems  to  have  been  cultivated  in  the 
minds  of  many  that  phenomena  connected  with  the  flow  of 
direct  electric  currents  and  of  alternating  electric  currents  are 
almost  entirely  unrelated.  This  popular  idea,  however,  is 
erroneous ; the  principles  which  relate  to  the  flow  of  electric 
currents,  whether  direct  or  alternating,  and  which  are  applied 
to  the  design  and  construction  of  machines  and  circuits,  are 
one  and  the  same. 

When  Oersted,  in  1820,  made  known  his  signal  discovery 
that  an  electric  current  exerts  a magnetic  influence  in  the 
space  around  it,  the  foundation  was  begun  for  our  knowledge 
of  the  laws  of  the  flow  of  alternating  currents.  Within  a 
dozen  or  fifteen  years  thereafter  much  knowledge  of  the  electric 
current  had  been  thrashed  out  experimentally  by  men  like 
Ampere,  Arago,  Faraday,  Henry,  and  others.  The  last  two 
named  laid  the  finishing  stone  on  the  foundation  by  searching 
out  and  making  known  the  laws  of  electro-magnetic  induction. 
These  basic  laws,  developed  early  in  the  nineteenth  century, 
apply  with  equal  force  to  continuous,  pulsating,  and  alternat- 
ing currents. 

In  dealing  with  alternating  currents,  several  variables  enter 
into  the  problem  which  make  it  impracticable  to  use,  without 
modification,  the  results  gained  in  the  study  of  direct  currents. 
But,  as  already  said,  the  same  fundamental  laws  control  the 
phenomena  of  both,  and  if  care  is  taken  to  apply  these  with 

1 


B 


o 


ALTERNATING  CURRENTS 


due  regard  to  tlie  limiting  conditions,  it  will  be  found  that 
the  subject  of  alternating  currents  may  be  clearly  grasped  and 
be  reduced  almost  to  the  simplicity  of  direct-current  work. 

The  design  and  operation  of  alternating-current  machinery 
may  differ  considerably  from  those  of  direct-current  apparatus 
in  certain  particulars  because  the  commercial  requirements  are 
materially  different,  but  the  same  principles  fundamentally 
apply  in  both,  and  in  the  matter  of  good  workmanship  and  sub- 
stantial construction  there  should  be  no  difference  in  the  two 
classes  of  machines. 

2.  Alternating  Voltage  and  Current. — An  alternating  voltage 
or  electric  pressure,  as  the  term  indicates,  is  an  uninterrupted, 
rapid  succession  of  electric  pressure  impulses  which  are  alter- 
nately in  opposite  di- 
rections, These  voltage 
impulses  produced  by  or- 
dinary commercial  alter- 
nating-current machines 
reverse  direction  many 
times  per  second  * and 
may  be  very  irregular  in 
form,  though  the  typical 
and  simplest  form  is  that 

Fig.  1.  — Two  Loops  of  a Sinusoidal  Alternating  0£  a splusoi(]-  Figure  1 
Voltage  or  Current  Curve.  3 

shows  the  plot  of  an  alter- 
nating voltage  curve  of  sinusoidal  form.  The  ordinates  of  the 
curve  represent  voltages,  while  the  abscissas  represent  time. 
The  abscissas  of  such  a curve  are  ordinarily  marked  in  degrees, 
or  in  fractions  of  7r;  the  base  of  the  two  loops  which  constitute 
a complete  cycle  of  a sine  wave  being  given  the  assigned  value 
of  360°,  or  2 7r  radians.  The  ordinates  may  be  scaled  in  volts. 
The  alternate  loops  are  drawn  above  and  below  the  zero  or 
datum  line  to  indicate  reversal  of  voltage. 

An  alternating  electric  current  flows  whenever  an  alternating 
voltage  is  applied  in  a closed  electric  circuit  of  fixed  constants. 
This  current  is  not  necessarily  proportional  at  each  instant  to 
the  voltage  ordinate  at  the  same  instant,  as  will  be  shown  later,  f 
A curve  representing  the  form  of  an  alternating  current  wave 
may  be  drawn  exactly  as  was  indicated  for  an  alternating  volt- 
* Art.  10.  t Chap.  IV. 


THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


3 


age,  but  the  ordinates  represent  amperes  instead  of  volts.  The 
simplest  typical  form  of  current  curve  is  also  sinusoidal,  but  this 
is  very  seldom  attained  in  commercial  practice;  while  on  the 
other  hand  highly  irregular  current  curves  are  quite  common. 

The  successive  loops  in  either  current  or  voltage  curves  are 
usually  assumed  to  be  of  identical  form  and  dimensions  as  long 
as  the  conditions  of  the  circuit  are  constant.* 

3.  Period  and  Frequency  of  an  Alternating  Current. — The 
time  in  seconds,  T \ required  to  pass  through  a complete  cycle 
(that  is,  two  loops  of  a curve)  is  called  the  Period  of  an  alter- 
nating electric  current  or  voltage.  The  number  of  periods  in  a 
second  is  called  the  Frequency  of  the  current  or  voltage  (this  term 
was  adopted  by  the  Paris  Electrical  Congress).  Usually  an  al- 
ternating current  or  voltage  is  designated  by  its  effective  f value 
and  frequency . It  is  common,  however,  to  use  the  number  of 
half -periods,  or  the  number  of  Alternations,  in  a minute,  instead 
of  the  frequency.  In  this  case,  the  number  of  alternations  is 
equal  to  2 x 60  x the  frequency.  Example  : a current  with  a 
frequency  of  60  periods  per  second  makes  7200  alternations  per 
minute.  The  term  Periodicity  is  sometimes  used  for  frequency. 

4.  Effective  Alternating  Voltages  and  Currents.  — The  heating 
effect  or  power  activity  of  a current  flowing  through  electrical 
resistance  varies  directly  as  the  square  of  the  current.  This 
was  proved  by  Joule  in  1841,  and  the  statement  is  often  called 
Joule's  law.  Putting  the  statement  of  the  law  in  symbols  gives 


where  H is  heat  measured  in  calories  ; /,  current  measured  in 
amperes  ; T,  the  time  measured  in  seconds  ; R , resistance 
measured  in  ohms  ; and  P is  the  power  expended  in  the  circuit, 
measured  in  watts. 

Alternating  currents  and  voltages  are  constantly  varying  in 
intensity,  and  the  heating  effect  of  an  alternating  current  is 
found  by  summing  up  instantaneous  values  through  a com- 
plete cycle,  as  follows  : 


in  which  i represents  instantaneous  values  of  current  ; and 
the  power  is 


. 24  I2R  T , and  the  power  is  P = I2R , 


* Art.  12. 


t Art.  4. 


4 


ALTERNATING  CURRENTS 


The  expression  vtX  Pdt  obviously  represents  the  average  of 


the  squared  instantaneous  currents , and  is  equal  to  the  average 
ordinate  of  a curve  plotted  on  the  same  base  as  the  original 
alternating-current  curve,  but  with  each  ordinate  of  a length 
equal  to  the  square  of  the  corresponding  ordinate  of  the  origi- 
nal curve. 

If  the  average  value  or  ordinate  of  an  alternating  current  is 

CT 

squared  (that  is,  [1/Ti  idt ]2),  the  result  is  not  the  same  as 


taking  the  average  of  squared  ordinates  or  instantaneous 
values ; and  the  amount  of  the  difference  depends  on  the 

form  of  the  current  curve. 
It  is  evident  that  this  must 
be  true,  since  squaring  the 
larger  instantaneous  values 
produces  more  than  a pro- 
portional effect  upon  the 
resulting  average.  It  is 
well  known  that  the  average 
of  the  squares  of  a series 
of  numerical  values  is 
always  larger  than  the 
square  of  the  average  of  the 
values,  unless  the  numeri- 
cal values  are  all  equal  to 
each  other. 

Figure  2 shows  a sine 
curve  and  its  curve  of  squared  instantaneous  ordinates  desig- 
nated respectively  A,  B , (7,  J),  E,  and  A , _P,  (7,  Q , E.  The 
height  of  the  line  EG  represents  the  average  value  of  the 
squared  current  ordinates,  while  the  height  of  the  line  A' T 
represents  the  square  of  the  average  of  the  current  ordinates. 

When  a sinusoidal  current  flows  through  electrical  resistance, 
the  average  power  expended  in  producing  heat  is  proportional  to 
the  average  of  the  squares  of  instantaneous  values  of  current, 
that  is,  to  the  height  of  the  line  EG  ; and  the  square  root  of  this 
ordinate  is  equal  in  amperes  to  a direct  current  which  (flowing 


P Q 


Fig.  2.  — Curve  of  Squared  Instantaneous 
Ordinates. 


T3E  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


b 


through  the  same  resistance)  will  have  the  same  heating  effect 
as  the  alternating  current  under  consideration.  This  square 
root  of  the  average  of  the  squared  instantaneous  values,  that  is, 
v/Av.  il  or  VAv.  e2,  is  called  the  Effective  value  of  the  current 
or  voltage. 

Proh.  1.  The  resistance  of  a circuit  is  20  ohms  and  500  watts 
are  expended  in  heating  it.  What  is  the  effective  current  in  the 
circuit? 

Proh.  2.  Four  circuits  in  parallel  and  having  1,  2,  4,  and  5 
ohms  resistance  respectively,  absorb  400,  200,  100,  and  80  watts. 
What  is  the  effective  current  in  each  branch  ? 

Prob.  3.  What  are  the  effective  voltages  across  the  terminals 
of  the  circuits  in  problems  1 and  2,  supposing  that  there  is  no 
live  counter- voltage  in  the  circuits? 

Prob.  4.  A circuit  of  10  ohms  resistance  is  in  series  with  the 
four  parallel  circuits  of  problem  2,  and  2000  watts  are  expended 
in  the  sj^stem.  What  are  the  effective  currents  and  voltages 
in  the  series  portion  and  in  the  parallel  portion  of  the  whole 
circuit,  supposing  no  live  counter-voltages  are  present  ? 

5.  Value  of  Effective  Current  and  Voltage  in  Terms  of  the 
Maximum  when  the  Curve  is  Sinusoidal.  — If  the  current  curve 
is  of  the  sine  form,  it  may  be  expressed  as  follows : 

i = im  sin  a, 

where  i is  the  instantaneous  value  of  the  current  for  any  angle 
a and  im  is  the  maximum  instantaneous  value.  To  obtain  the 
average  heating  effect  of  this  current  when  flowing  through 
a resistance,  square  both  sides  of  the  equation,  multiply  through 
by  i?,  and  find  by  integration  the  area  of  one  loop  or  one-half 
period  of  the  squared  curve.  The  area  thus  found  divided  by 
the  base  nr  gives  the  average  of  the  squared  ordinates.  The 
expression  may  conveniently  take  the  form, 

(Av.  «2)i?  = — i}n  P sin2  ad  a, 

7 T *A) 

7?;  2 

from  which,  ( Av.  i2)  R = — -• 


G 


ALTERNATING  CURRENTS 


Dividing  by  R,  Av. 


9*2 


or 


V Av.  i2  — ~~=  — A07*„ 

V2 


The  value  of  VAv.  i2  is  called  the  Effective  value  of  the 
alternating  current,  as  has  already  been  pointed  out. 

The  effective  value  of  an  alternating  current  is  the  square  root 
of  the  average  of  the  squared  instantaneous  values  of  the  current. 
This  is  correct  whether  or  not  the  current  is  of  sinusoidal  form. 
The  foregoing  equations  show  that,  when  the  current  is  sinu- 
soidal in  form,  there  is  a fixed  relation  between  the  effective 
and  maximum  values  of  the  current  and  that  the  effective 

value  is  .707  ( = — ] times  the  maximum  value. 

V V2  J 

In  the  case  of  a sinusoidal  pressure  (voltage)  curve  the  for- 
mula may  be  written, 

e - em  sin  «, 

where  e is  the  instantaneous  value  of  the  voltage  for  any  angle  a 
and  em  is  the  maximum  instantaneous  value.  By  writing  the 
integral  for  the  heating  effect  as  before,  a similar  result  is  ob- 
tained, or 

Av.  e2  _ Mj 
2 R 


and 


R 

v/Av.  e2  = 


V2 


= -707  em. 


Rffective  voltage  is  the  square  root  of  the  average  of  the  squared 
instantaneous  values  of  the  voltage.  When  the  form  of  the 
voltage  wave  is  sinusoidal,  the  effective  voltage  is  equal  to 


.707  f = times  the  maximum  value  of  the  voltage. 

V V2 / 

The  average  value  of  the  ordinates  of  a sine  curve  may  be 
found  from  the  expression 


Av.  y = j sin  ada, 
7 r 


where  y is  the  instantaneous  ordinate  and  ym  the  maximum 
ordinate  of  the  curve  ; and  therefore 


Av.  y = — .037 


y mi 


THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


7 


2 

or  the  average  value  of  the  instantaneous  ordinates  is  — ( = .637) 

7T 

times  the  maximum  ordinate.  From  this  it  is  seen  that 
VAv.  y1  — (Av.  «/), 

or  VAv.  ?/2  = 1.11  (Av.  y'). 

Therefore  the  ratio  of  effective  current  or  voltage  to  the  average 
ordinate  of  the  curve,  in  the  case  of  sinusoidal  curves,  has  the 
fixed  value  of  1.11. 

The  same  considerations  follow  for  alternating  current  and 
voltage  waves  of  any  form,  but  the  ratios  of  the  average  and 
the  effective  values  to  the  maximum  are  different,  and  depend 
upon  the  form  of  each  curve  considered.  Alternating  current 
and  voltage  waves  are  always  single-valued  ; that  is,  a single 
value  of  the  ordinates  corresponds  to  each  single  value  of  the 
abscissas.  They  may  be  represented  by  the  expression 

x=af(u~). 

Manifestly,  the  ratio  which  the  average  bears  to  the  maximum 
ordinate  takes  a value  nearer  unity  for  a curve  that  is  flatter 
than  a sinusoid,  and  recedes  farther  from  unity  for  a curve  more 
peaked  than  a sinusoid. 

For  a curve  made  up  of  alternating  rectangular  loops,  such 
as  is  shown  in  Fig.  17,  the  maximum,  average,  and  effective 
values  are  equal  to  each  other ; but  for  a curve  made  up  of 
alternating  triangular  (isosceles)  loops,  the  average  value  is 
one  half  the  maximum  value,  and  the  effective  value  exceeds 
the  average  value  in  a ratio  of  1.155  to  1.  The  ratio  of  the 
effective  to  the  average  value  may  be  quite  large  in  the  case  of 
a curve  which  is  very  peaked.  This  ratio  is  sometimes  called 
the  Form  factor  of  the  curve. 

Prob.  1.  What  are  the  maximum  ordinates  of  the  sinusoidal 
voltages  having  effective  values  of  100  and  200  volts,  and  of 
sinusoidal  currents  having  the  effective  values  of  50  and  75 
amperes  ? 

Prob.  2.  What  are  the  average  ordinates  of  voltages  and 
currents  in  problem  1? 

Prob.  3.  What  are  the  effective  and  average  values  of  a 


8 


ALTERNATING  CUR  1 i ENTS 


voltage  wave  which  is  of  the  form  of  an  equilateral  triangle, 
and  lias  an  altitude  of  110  volts? 

Prob.  4.  What  are  the  effective  and  average  values  of  a 
voltage  wave  in  the  form  of  a right  triangle  in  which  the  alti- 
tude is  10  and  the  base  10? 

Prob.  5.  What  are  the  average  and  effective  values  of  a cur- 
rent wave  having  the  form  of  a semicircle  with  the  base  equal 
to  100?  That  is,  compute  the  average  ordinate  and  the  square 
root  of  the  average  of  the  squared  ordinates,  in  terms  of  the 
radius. 

6.  Vectors  representing  Sinusoidal  or  Harmonic  Voltages  and 
Currents ; Lead  and  Lag- — A physical  quantity  which  may  be 
represented  by  the  length,  position,  and  direction  of  a line  is 
called  a Vector  quantity  and  the  line  so  representing  the 
quantity  is  called  a Vector.*  Thus,  two  forces  may  be  repre- 
sented (1)  in  magnitude,  sometimes  termed  Scalar  value,  by  two 
lines  having  lengths  proportional  to  the  intensities  of  the 
forces  ; (2)  in  relative  angular  position,  or  Phase,  by  the  angle  at 
which  the  lines,  extended  if  necessary,  intersect ; and  (3)  in  di- 
rection by  arrow  heads  placed  upon  the  lines.  In  the  following 
discussions,  if  one  or  more  vectors  radiate  from  or  turn  upon  a 
common  center,  and  no  arrow  heads  are  shown,  it  is  assumed 
that  their  directions  are  from  the  center  outward.  Vectors 
may  be  combined  or  resolved  into  components  by  the  well-known 
laws  of  mechanics  relating  to  the  composition  or  resolution  of 
forces  or  velocities.  Thus,  the  centrifugal  force  acting  upon  a 
uniformly  rotating  crank  may  be  represented  by  a vector  rotat- 
ing uniformly  around  one  end  as  a center  of  revolution , and  its 
component  value  at  any  instant  in  the  direction  of  some  fixed 
line  (assume  for  convenience  a vertical  line  passing  through 
the  axis  of  rotation)  is  found  by  projecting  the  vector  from 
its  position  at  the  instant,  upon  that  line.  If  the  center  is 
moved  (for  convenience,  horizontally  to  the  right)  a given  dis- 
tance for  each  degree  of  angular  rotation  of  the  vector,  and 
the  instantaneous  vertical  projections  for  each  degree  of  ad- 
vance are  plotted  at  the  proper  points  along  the  path  of  the 
center,  the  points  thus  obtained  when  joined  form  a sine  curve. 
In  Fig.  3,  OA  is  a rotating  vector.  OX  is  the  initial  line  or 

-*  For  a discussion  of  the  use  of  vectors  in  graphical  methods  of  solving  elec- 
trical problems,  see  Chapter  VI. 


THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


9 


horizontal  axis  from  which  the  angle  a is  measured,  OY  is  the 
vertical  axis  upon  which  the  point  A is  projected  as  the  vector- 
rotates,  and  SS  is  a 
sine  curve  produced 
as  explained  above. 

In  the  figure,  OA  has 
advanced  30°,  and 
the  ordinate  a'  of  the 
sine  curve,  which  is 

placed  over  the  30  Produced  Thereby, 

division  of  its  base, 

is  made  equal  to  OA  sin  30°  or  Oy.  Likewise,  when  the  vector 
has  advanced  to  60°  the  ordinate  a"  of  the  curve  is  equal  to 
OA  sin  60°.  When  the  vector  is  at  90°,  the  ordinate  a’"  is 
equal  to  OA  sin  90°,  which  equals  OA\  hence  the  length  of  a 
rotating  vector  required  to  produce  a given  sine  curve  must  be 
equal  to  the  maximum  ordinate  of  the  curve. 

By  geometry  it  is  seen  that  in  the  figure 

A = Vy2  + x2  = A (sin2«  + cos2a)^, 
where  A = OA,  x = Ox  — A cos  a,  y = Oy  = A sin  a ; and 


tan  a = ^ and  tan 
x 


x 


These  relations  express  the  angular  position  and  value  of  a 
Rotating  vector  in  terms  of  its  instantaneous  rectangular  com- 
ponents with  great  simplicity,  and  are  much  used  in  the  compu- 
tations of  electrical  phenomena. 

Rotating  vectors  may  be  combined,  by  the  ordinary  methods 
of  mechanics,  with  other  vectors  having  the  same  speed  of 
rotation,  since  their  relative  angular  positions  will  remain 
always  the  same.  Thus,  in  Fig.  4,  OA  and  OB  are  two  rotat- 
ing vectors  having  the  same  speed  or  frequency,  and  therefore 
at  a fixed  Phase  difference  or  angular  position  with  respect  to 
each  other.  Their  resultant  is  OO,  the  diagonal  of  the  paral- 
lelogram OACB.  It  may  be  seen  by  inspection  that  the  ver- 
tical projection  of  OC,  which  is  Oc , is  equal  at  any  instant  to 
A sin  a + B sin  (u-0)=Oa- f Ob,  since  BO  is  equal  and  par- 
allel to  OA  and  ac  is  therefore  equal  to  Ob.  These  vectors 
A,  B,  and  0 are  the  generators  of  the  sine  curves  or  harmonics 


10 


ALTERNATING  CURRENTS 


A',  B' , and  C . If  the  ordinates  of  A'  and  B'  corresponding 
to  any  abscissa  are  added  together,  they  give  the  corresponding 


Fig.  4. — Composition  of  Rotating  Vectors  OA  and  OB  by  Parallelogram  of  Forces. 


ordinate  of  the  curve  C' . This  follows  from  the  construction 
of  the  curves.  If  the  vector  OC  is  given,  it  can  be  resolved 

into  the  component  vectors  OA 
and  OB  with  their  particular  phase 
relations  to  OC , or  it  may  be  re- 
solved into  components  at  any 
other  fixed  angular  relations,  by 
reversing  the  process  of  construc- 
tion. Or,  if  OC  and  one  compo- 
nent, OA,  are  given,  the  other 
component  may  be  found  by  com- 
pleting the  parallelogram  of  which 
OC  is  the  diagonal  and  OA  one 
side.  Three  or  more  vectors  may 
be  combined  by  similar  processes. 

In  the  case  of  two  vectors  90°  apart  the  relations  are  very 
simple.  Thus  in  Fig.  5,  if  A and  B are  two  such  vectors,  rotat- 
ing about  Z,  which  form  the  resultant  C, 

B 


Fig.  5. — Illustration  of  the  Relations 
of  Two  Vectors  90°  apart. 


C=  V^L2  + B'2,  tan  6 = —,  tan  6'  = 

A -B 

a = A sin  a,  and  b = B sin  («  -f-  90°)  = B cos  «. 

where  A,  B , and  C are  the  values  of  the  vectors,  a and  b are 
the  vertical  components  of  A and  B.  a is  the  angle  of  advance 
of  A,  and  6 and  6'  are  the  fixed  angles  AZC  and  CZB.  Further, 

c—C  sin  («  + 6), 

and  c = a + b. 


THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


11 


where  a,  b , and  c are  the  instantaneous  vertical  components  of 
A,  B,  and  C;  from  which, remembering  that  cos  « = sin  (a  + 90°), 

c = A sin  u + B cos  « ; 

hence,  A sin  « + B cos  a = (7 sin  («  + 0). 

Also  A sin  a + B cos  a — C cos  («  — 6 

In  using  these  expressions  it  is  desirable  to  always  measure  a 
in  the  positive  (counter-clockwise)  direction  in  order  to  pre- 
vent confusion  in  the  algebraic  signs  of  the  angles. 

Since  B cos  a = B sin  (« + 90°),  it  is  evident  that  a sine 
curve  generated  by  a rotating  vector  90°  in  advance  of  the 
vector  OA  of  Fig.  3 would  be  a curve  with  its  positive  maxi- 
mum at  a = 0°,  its  negative  maximum  at  a = 180°,  and  its  zero 
values  at  a = 90°,  270°,  etc.  That  is,  this  cosine  curve  would 
be  ahead  of  or  in  the  Lead  of  the  curve  drawn  in  Fig.  3 by  90°, 
and  vice  versa , the  sine  function  Lags  90°  behind  the  cosine 
function.  Also,  the  phase  or  relative  angular  position  of  a re- 
sultant vector  is  between  the  phases  of  its  components,  and  the 
angles  between  a resultant  and  its  components  depend  upon 
the  phase  difference  and  the  ratio  of  scalar  values  of  the  com- 
ponents. 

Vectors  having  different  speeds  of  rotation  cannot  be  com- 
bined by  the  parallelogram  of  forces,  as  the  resultant  vector 
varies  in  length  with  the  time.* 

Harmonic  (sinusoidal)  voltages  and  currents  can  evidently 
be  represented  by  rotating  vectors  and  can  be  combined  with 
as  much  facility  as  mechanical  forces  or  velocities.  In  doing 
this'the  maximum  ordinate,  or  Amplitude,  must  be  taken  for  the 
scalar  value  of  the  vector  if  it  is  desired  to  determine  the  in- 
stantaneous components ; but  if  it  is  desired  to  find  the  effec- 
tive resultant  of  two  or  more  voltages  or  currents,  the  effective 
values  of  the  components  may  be  used.  This  is  because 

E = : and  therefore  the  parallelograms  using  maximum  and 

effective  values  are  similar,  the  former  being  convertible  into 
the  latter  by  dividing  its  sides  and  diagonals  by  V2.  Effective 
voltages  and  currents  may  possibly  not  be  represented  as  vec- 
tors in  the  broadest  mathematical  sense,  but  for  our  purposes 
such  representation  is  eminently  satisfactory. 

* See  Art.  71. 


12 


ALTERNATING  CURRENTS 


K"/ 


If  a number  of  vectors  are  to  be  combined,  the  resultant  of  a 
pair  may  be  found  and  this  then  combined  with  another  vector, 

and  so  on  until  the  final  result- 
ant is  obtained.  In  Fig.  6 the 
resultant  of  the  vectors  A,  A', 
A",  and  A'"  is  found  by  the 
parallelograms  OAa'A', 
Oa'a"A",  and  Oa"a"'A'". 
This  is  cumbersome  and  can 
be  simplified  by  drawing  OA , 
Aa' , a' a",  and  a" a'",  respec- 
tively, parallel  and  equal  in 
x length  to  the  vectors  A , A', 

Fig.  6.  Vector  Addition  by  Parallelo-  i A"1  Bv  ioillino-  the 

grams  of  Forces  and  Vector  Polygon.  . ‘ ' 

free  points  O and  a"  the  re- 
sultant OR  is  obtained.  The  figure  OAa'a" a"'  0 is  called  a 
Vector  polygon.  If  only  two  vectors  are  combined,  giving,  say, 
OAa' , the  result  is  called  a Vector  triangle.  When  the  figure 
shows  all  the  vectors  radiating  from  a center  as  do  OA,  OA', 
OA",  and  OA'",  it  is  called  a Phase  diagram,  as  the  relative 
angular  positions  of  the  vectors  are  directly  indicated. 


Prob.  1.  Two  current  vectors  differing  30°  in  phase  have 
values  respectively  of  20  and  40  amperes.  What  is  their  com- 
bined value  ? 

Prob.  2.  Four  voltages  of  25,  50,  75,  and  100  are  represented 
by  vectors  having  0°,  30°,  60°,  and  90°  absolute  phase  positions. 
What  is  the  combined  value  of  all  the  voltages  in  series,  and 
what  is  its  angle  with  reference  to  the  component  vector  hav- 
ing 0°  angle  ? 

Prob.  3.  A current  of  100  amperes  divides  into  two  branches 
in  such  a manner  that  one  branch  current  leads  the  main  cur- 
rent by  30°,  and  the  second  branch  current  lags  behind  the 
main  current  by  an  angle  of  60°.  What  are  the  values  of  the 
two  currents  ? 

Prob.  4.  A current  vector  of  50  amperes  is  divided  into  two 
rectangular  components,  one  of  which  lags  behind  it  by  an 
angle  of  45°.  What  is  the  value  of  the  second  component  and 
its  angular  position  ? 


THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


13 


Prob.  5.  A voltage  vector  of  100  volts  is  composed  of  two 
components,  one  of  which  lags  behind  it  by  an  angle  of  10°, 
and  the  other  leads  by  an  angle  of  20°.  What  are  the  scalar 
values  of  the  components  ? 

Prob.  6.  There  are  four  voltages,  A,  B , (7,  i),  in  series  in  a 
circuit.  Assuming  the  first  to  have  an  angle  of  0°  and  the  others 
to  be  measured  in  angular  positions  with  reference  to  it,  the 
scalar  and  angular  values  of  the  vectors  are  10,  A 0°  ; 20,  Z 15° ; 
50,  Z — 15° ; 30,  Z — 45°.  What  is  the  resultant  voltage  and 
its  angular  position  with  reference  to  the  component  having 
Z0°? 

Prob.  7.  The  resultant  in  problem  6 is  to  be  divided  into 
two  rectangular  components,  one  on  the  horizontal  axis  (Z  0°), 
and  the  other  on  the  vertical  axis.  What  are  the  scalar  values 
of  the  components? 


7.  Polar  Coordinates  and  the  Method  of  obtaining  Effective 
Voltages  from  the  Polar  Curve.  — It  is  sometimes  convenient  to 
plot  alternating  voltage  and  current  curves  to  polar  coordi- 
nates ; i.e.  the  instantaneous  values  of  the  voltage  or  current 
are  laid  off  on  radius  vectors  occupying  the  corresponding  in- 
stantaneous positions  of  the  rotating  vector  thought  of  as  gen- 
erating the  alternating  function. 

The  effective  value  may  be  directly  derived  from  the  primary 
polar  curve,  as  originally  shown 
by  Steinmetz,*  if  it  is  plotted  on 
polar  coordinates  taking  360°  to 
a complete  period.  This  gives  a 
symmetrical  curve  which  crosses 
the  origin  at  0°,  180°,  360°,  etc. 

For  an  exact  sinusoid  the  curve  is 
of  the  form  shown  in  Fig.  7,  and 
has  its  maximum  values,  positive 
and  negative,  at  90°  and  270°; 
i.e.  each  loop  is  a circle  with  the 
pole  on  its  circumference  and  the 
initial  line,  C*A,  tangent  to  the  Fig.  7. — Polar  Diagram  of  a Harmonic 
circumference,  the  maximum  Function. 

* Trans.  Amer.  Inst.  E.  E.,  Yol.  10,  p.  527  ; Elektrotechnische  Zeitschrift , 
June  20,  1890. 


14 


ALTERNATING  CURRENTS 


ordinate,  a , being  equal  to  the  diameter.  The  area  of  the  curve 
in  this  form  may  be  shown  to  be  directly  proportional  to  the 
square  of  the  effective  value  of  the  ordinates  as  follows : In 
the  case  of  a sinusoidal  curve,  the  polar  curve  has  the  equation 
e = a sin  a,  where  e is  the  instantaneous  voltage  corresponding 
to  an  angular  advance  a.  In  plotting  the  curve,  values  of  e are 
laid  off  on  the  radius  vectors  having  vectorial  angles  equal  to 
the  corresponding  values  of  a,  and  a line  is  drawn  through 
the  points  thus  located  (Fig.  7).  Each  loop  of  this  curve,  that 
is,  the  part  of  the  curve  taken  between  « = 0°  and  « = 180°,  or 
a — 180°  and  a = 360°,  is  a circle,  and  its  area  is  A = ^ ird2, 
where  d is  the  diameter  of  the  circle.  By  the  construction,  d 
is  equal  to  a of  the  formula  e = a sin  «,  and  the  area  of  a loop 
of  the  curve  is  therefore  A = \ira2.  The  effective  ordinate  of 
a sinusoid  has  already  been  shown  to  be  * 


a 

V2 


Consequently 


= .798  VA. 

' 7 r 


This  may  be  taken  for  most  purposes  as  E = .8VA. 

The  same  expression  applies  to  any  single-valued  periodic 
curve  of  equal  positive  and  negative  loops.  The  area  of  one 
loop  of  any  such  polar  curve  is 

A = r e2da  ; 

for  if  the  curve  is  divided  up  into  elementary  triangles  having 
their  apices  at  0 and  bases  equal  to  eA «,  each  triangle  has  an 
area  approximately  equal  to  A « x e2,  where  e is  the  average 
altitude  of  the  triangle;  and  the  total  area  is 

A = e2A a. 

^0- 

This,  when  Aa  is  reduced  to  the  limit,  becomes  the  integral 
given  above.  The  mean  of  the  squared  ordinates  of  the  func- 
tion is  Av.  e2  — — f e2da.  The  effective  ordinate  is  therefore 

TT'-'O 

E — VAv.«2=  \/—  f e2du  — \/:^  = .798  Vi. 

' 7r*A>  ' 7 r 

As  before,  this  may  be  taken  as  E = .8 VA. 

* Art.  5. 


THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


15 


Figure  8 shows  the  voltage  curve  plotted  to  rectangular 
coordinates,  the  curve  of  squared  ordinates,  and  the  polar  curve 
for  the  voltage  wave  of  an 
alternator  which  bj  special 
design  gave  a very  distorted 
wave.  The  mean  ordinates 
of  the  rectangular  curves 
are  readily  determined  by 
measuring  the  area  by  plani- 
meter  and  dividing  the  area 
by  the  length  of  the  base. 


Prob.  1.  Construct  the 
polar  loops  of  a wave  of 
voltage  which,  laid  out  to 
rectangular  coordinates,  has 
rectangular  loops  with  a 
base  of  10  and  an  altitude 
of  5. 

Prob.  2.  Make  the  same 
construction  as  in  problem 
1,  when  the  wave  plotted  on 
rectangular  axes  is  a semi- 
circle having  a diameter  of  8. 

Prob.  3.  Find  the  effec- 


Fig. 8. — Irregular  Alternator  Voltage  Loop 
plotted  to  Rectangular  and  Polar  Coordi- 
nates, and  the  Curve  of  Squared  Ordinates. 


tive  values  of  the  voltages  in  problems  1 and  2 by  the  method 
explained  in  this  article. 


8.  A Fundamental  Conception  of  the  Vector  Quantity.  — In 
Art.  6 it  has  been  pointed  out  that  currents  and  voltages 
may  be  shown  in  magnitude  and  relative  phase  by  means  of  a 
phase  diagram.  Thus,  in  Fig.  9 suppose  OX  to  be  the  initial 
line  and  OA',  OA ",  and  OA !"  to  be  voltages  in  series,  or  cur- 
rents entering  or  leaving  a junction,  which  are  represented  in 
relative  phase  by  the  angular  positions  and  in  magnitude  by 
the  lengths  of  the  lines.  It  has  been  pointed  out  that  the 
resultant  of  two  or  more  similar  electrical  quantities  may  be 
found  by  treating  their  representative  lines  as  vectors ; such 
vectors  may  be  combined  by  algebraically  adding  the  vertical 
and  horizontal  components  of  the  individual  lines,  by  which 


1G 


ALTERNATING  CURRENTS 


means  the  vertical  and  horizontal  components  of  the  resultant 
are  determined.  If  a',  V ; a",  b" ; and  a b'"  are  the  hori- 
zontal and  vertical  components  of  OA',  OA",  and  OA'",  and 
A,  B,  the  components  of  the  resultant,  then 


or 


A + B = (V  + a"  + a'"')  + (V  + b"  + 5'"), 

A + B = O'  + 5')  + ( a " + b")  + (ct!"  + V"). 


W 

o 

1'lvi 


X/! ' 

\ 

\ 

\ 

\\ 

A 

rtt  s' 

s' 

/ / ! 

/ 1 

r / 

! 

/ " 
J A 

:7^P 

/,X''Z 

O'  / 

i i 

'r  1 1 

i 

9.  — Illustration  of  Relation  of  Vectors 
and  tlieir  Components. 


In  this  expression  there  is 
nothing  to  distinguish  the  hor- 
izontal from  the  vertical  com- 
ponents except  the  difference 
between  the  letters  A or  a and 
the  letters  B or  b ; or,  in  gen- 
eral, there  is  nothing  to  indi- 
cate the  angular  positions  of 
the  components,  or  of  the  lines 
represented  by  them,  with 
reference  to  the  initial  line ; 
and  the  ordinary  algebraic 
processes  afford  no  convenient 
means  for  giving  these  indica- 
tions. To  fully  indicate  the  magnitude  and  position  of  a line 
by  its  rectangular  components,  we  must  abandon  the  methods 
of  algebra  for  geometric  processes.  Therefore  we  may  con- 
sider, for  the  moment,  that  tire  compo- 
nents t and  u of  the  vector  A (Fig.  10) 
both  lie  on  the  initial  line  OX , but  in 
order  that  t and  u may  determine  the 
vector  A,  u must  be  vertical,  that  is, 
rotated  90°.*  To  indicate  such  a ro- 
tation, a prefix  such  as  j may  be  used. 

Then  A will  be  represented  in  mag- 
nitude and  angular  position  by  the  Fig 
expression 


t x 

10.  — Single  Vector  and 
Components. 


A=  t +ju, 


where  the  sole  function  of  the  letter  j is  to  indicate  that  the 
component  u stands  90°  from  the  initial  line  and  the  addition  is 

* The  letter  denoting  a vector  will  be  written  with  a vinculum,  thus,  A,  to  dis- 
tinguish it  from  its  scalar  value.  When  its  components  are  on  the  horizontal  or 
vertical  they  will  have  this  designation  omitted. 


THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


17 


geometric;  that  is,  the  joint  effect  of  two  vectors  in  the  direction  of 
their  resultant  is  equal  to  the  effect  of  their  resultant.  The  effect 
of  t and  ju  is  equal  to  the  effect  of  A.  The  expression  t A ju 
does  not  indicate 
arithmetical  addition, 
hut  represents  the 
combination  of  the  ef- 
fects of  t and  u.  It 
is  sometimes  written 
t , ju.  As  ju  is  posi- 
tive, it  is  said  to  have 
rotated  u ahead  90°; 

— ju  would  indicate 
that  u had  been  ro- 
tated 90°  in  a nega- 
tive direction,  or  that 
u is  measured  down- 
wards (the  negative 

direction)  from  the 

. . 1 . . Fig.  11.  — Illustration  of  the  Effect  of  the  Operator  j. 

origin.  It  t and  ju 

are  both  negative,  they  are  both  measured  in  the  negative 
direction ; hence,  if  t +ju  be  multiplied  by  — 1,  there  results 

— t — ju , t and  u are  both  reversed  in  direction,  and  the  vector 
line  OA  is  rotated  180°  (Fig.  11);  jt  — u means  that  the  line 
has  been  rotated  forward  90°,  since  t is  positive  but  stands  at 
90°  from  the  initial  line,  and  u is  negative  ; — jt  + u means  that 
the  line  has  been  rotated  back  90°.  Finally,  multiplying  by/ 
means  advancing  the  vector  line  90°;  for,  j(t  + ju)  =jt  -f  (+/) 
(/m),  and  as  j2  indicates  rotation  twice  forward,  j2u  becomes 

— u,  and  therefore  j (t  + jii)  is  geometrically  equal  to  jt  — u ; j 
is,  therefore,  seen  to  be  similar  to  the  imaginary  term  V — 1 
since  (V—  1)2=  — 1.  Also  multiplying  by  —j  means  turning 
the  vector  back  90°,  for  — j(t  + ju)=  —jt  + (—/)(/«),  and  as 

— j2  (namely,  +/  x — /)  indicates  rotation  forward  90°  and 
back  90°,  — j2u  = u , and  there  results  — jt  + u. 

Quantities  comprising  associations  of  real  and  imaginary 
quantities,  such  as  the  quantities  described  in  the  preceding 
paragraph,  are  called  Complex  quantities. 

The  vector  expressing  a sine  wave  may  now  be  represented 
in  magnitude  or  scalar  value,  as  heretofore  shown,  by 


18 


ALTERNATING  CURRENTS 


A = Vt2  + u2, 

where  A is  the  length  or  scalar  value  of  the  vector  A and  is  a 
purely  arithmetical  quantity;  in  phase  by 

tan  6 = - ; 
t 

in  phase  and  magnitude  by  the  complex  quantity 
aL(cos  6 +j  sin  6), 

since  A cos  8 — t and  A sin  6—  u ; and  also,  as  just  indicated, 
by  the  equivalent  complex  quantity 

t +ju. 

The  addition  of  the  vectors  given  in  the  first  illustration 
of  this  article  now  becomes 

A +jB  = ( a ' + a"  + a'"')  + j(b'  + b"  + b'"). 

The  assumptions  made  with  respect  to  the  symbol  j leave  it 
subject  to  the  fundamental  laws  of  algebra,  even  though  it 
does  not  represent  an  ordinary  numerical  quantity  and  may  be 
considered  purely  as  a sign  of  operation. 

For  a further  discussion  of  the  complex  quantity  and  its 
application  to  various  types  of  electric  circuits,  refer  to 
Chapter  VI. 

Prob.  1.  The  complex  quantity  expressing  a vector  of  vol- 
tage is  10  + /8.  What  is  the  angular  position  of  this  vector 
with  reference  to  the  horizontal  axis,  Z 0°,  and  what  is  its 
scalar  value  in  volts? 

Prob.  2.  A vector  of  current  is  1=  15  — j 6.  What  is  its 
angular  position  and  scalar  value? 

Prob.  3.  Three  vectors  of  voltage  Ex  = 5 +j  6,  = 8 + 

j 10,  Ez  = 20  +/  2 are  to  be  added  together.  What  is  the  angu- 
lar position  and  scalar  value  of  the  resultant  voltage  ? 

Prob.  4.  Complex  quantities  expressing  three  voltages  which 
are  in  series  are  Ex  = 80  — j 10,  E2  = 100  +j  0 and  Ez  = —j  50. 
What  is  the  scalar  value  and  angular  position  of  the  resultant 
voltage? 

Prob.  5.  A current  having  a scalar  value  of  100  amperes 
and  an  angular  position  of  Z 30°  may  be  expressed  as  a com- 
plex quantity  in  what  terms  ? 


TIIE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


19 


Prob.  6.  Two  vectors  of  voltage  having  respectively  scalar 
values  of  20  and  50  and  angles  of  Z 30°  and  Z — 30°  are  to  be 
combined.  Form  their  complex  expressions  and  from  these 
obtain  the  complex  expression  of  the  resultant,  after  which 
find  the  scalar  value  and  angular  position  of  the  resultant. 

Prob.  7.  Find  the  scalar  value  and  angular  position  of  the 
resultant  of  the  four  following  voltages  : 

-#i  = 8 +j  4,  J?2  = 10  -j  6,  Jj3  = 15,  Ei  =j  20. 


Fig.  12.  — Simple  Alternator  Diagram. 


9.  The  Principle  of  an  Alternating-current  Generator.  — The 
simplest  form  of  Alternating-current  generator,  or  Alternator,  is 

a single  coil  revolving  in  a constant  magnetic  field  and  having 
its  terminals  connected  to  two  rings  fastened  upon  the  shaft. 
Such  a simple  ar- 
rangement is  shown 
diagrammatically  in 
Fig.  12.  The  rings 
to  which  the  ends 
of  the  coil  are  at- 
tached are  called 
Collector  rings  or 
Slip  rings,  and  upon 
these  bear  copper  or 
carbon  brushes  for 
the  purpose  of  connecting  the  coil  with  the  external  circuit. 
Revolving  the  coil  in  the  magnetic  field  causes  a voltage  to  be 
generated  in  one  direction  during  the  time  it  is  under  the  influ- 
ence of  one  pole  piece,  and  the  direction  of  the  voltage  reverses 
as  soon  as  the  coil  begins  cutting  lines  in  the  opposite  direction 
under  the  influence  of  the  other  pole.  If  the  instantaneous 
voltages  set  up  in  the  revolving  coil  are  plotted  as  explained 
in  Art.  2,  a curve  of  two  loops,  or  one  complete  period,  will 
represent  the  results  in  each  complete  revolution  (Fig.  1). 
As  an  alternating  current  will  not  serve  to  magnetize  the  fields 
of  alternating-current  generators  of  this  type,  which 'are  usually 
called  Synchronous  generators,  some  special  arrangement  for 
obtaining  a direct  current  for  the  excitation  must  be  made. 
This  may  be  done  either  by  commutating  or  rectifying  all  or 
a part  of  the  alternating  current  produced  by  the  machine,  or 


20 


ALTERNATING  CURRENTS 


a small  auxiliary  direct-current  dynamo  called  an  Exciter  may 
be  supplied  for  the  purpose.  Sometimes  the  exciter  is 
mounted  on  the  bed  plate  of  the  alternator. 

10.  Commercial  Frequencies  and  Multipolar  Generators. — . 
The  frequencies  of  alternating  currents  used  for  the  general 
commercial  purposes  of  the  present  day  vary  widely,  but  in 
nearly  all  cases  fall  within  the  limits  of  15  and  135  periods  or 
Cycles  per  second  (1800  and  16,200  alternations  per  minute). 
The  majority  of  American  alternating-current  dynamos,  or 
Alternators,  give  a frequency  of  25,  40,  and  60  periods  per 
second. 

The  rotation  of  a coil  in  a two-pole  field  gives  one  complete 
period,  or  two  alternations,  for  each  revolution ; and  the  fre- 
quency is  therefore  equal  to  the  number  of  revolutions  per 
second.  Since  the  armatures  of  two-pole  machines,  unless 
driven  by  direct-connected  steam  turbines  or  the  like,  would 
be  required  to  run  at  impracticable  speeds  in  order  to  give 
the  ordinary  commercial  frequencies,  alternators  are  usuallv 
made  with  a considerable  number  of  poles.  The  number  of 
poles  depends  upon  the  size  of  the  alternator  and  other  condi- 
tions which  may  control  the  speed  of  the  armature,  but,  in 
general,  it  may  be  said  to  vary  from  eight  upwards.  An  ex- 
ception to  this  statement  must  be  made  in  the  case  of  generators 
which  are  designed  for  direct  connection  with  steam  turbines. 
The  unusually  high  speed  of  these  turbines  renders  a small 
number  of  field  poles,  even  as  few  as  two  or  four,  requisite  in 
the  direct-connected  alternator. 

Figure  13  is  a diagram  of  a six-pole  generator  with  revolving 
armature.  In  the  figure  A7",  S,  JY,  S,  JY,  S,  are  the  pole  pieces, 
Y is  the  yoke,  A the  armature,  B the  brushes,  and  TF  TF  the 
field  windings.  Two  of  the  magnetic  circuits  of  the  machine 
are  shown  by  the  dotted  lines  marked  n , n.  Instead  of  using 
one  coil  on  the  armature  as  was  shown  in  Fig.  12,  it  is  usual  in 
this  type  of  machine  to  use  as  many  as  there  are  poles  for  the 
purpose  of  utilizing  the  working  space  to  the  best  advantage. 
These  coil's  may  be  conveniently  connected  in  series  and  their 
free  ends  connected  to  the  collector  rings,  as  is  shown  at  B, 
Fig.  13,  but  they  may  be  connected  in  series  parallel  or  parallel 
relations  if  desired.  In  the  series  connection,  the  alternate 
coils,  which  move  under  pole  pieces  of  opposite  polarities,  must 


THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


21 


be  connected  in  circuit  in  relatively  opposite  directions  so  that 
their  voltages  may  add  together. 

The  frequency  which  is  produced  by  an  alternator  of  this 
type  is  equal  to  one  sixtieth  of 
the  product  of  the  number  of 
its  pairs  of  poles  by  the  number 
of  revolutions  per  minute  made 
by  the  armature.  It  is  evident 
that  this  must  be  the  case,  since 
an  armature  coil  will  have  com- 
pleted one  cycle  or  period  when 
its  center  has  passed  from  a 
point  under  one  magnetic  pole 
to  a similar  position  with  re- 
spect to  the  next  pole  of  the 
same  sign.  The  number  of  al- 
ternations per  minute  is  equal  to  the  frequency,  as  shown 
above,  multiplied  by  120. 

In  any  alternator,  one  complete  cycle  of  the  alternating  vol- 
tage is  generated  while  an  armature  conductor  moves  from  a 
given  point  under  a pole  of  one  polarity  to  a like  point  under  the 
next  pole  of  the  same  polarity ; and  the  frequency  is  equal  to  the 
number  of  cycles  per  second.  This  definition  applies  to  all 
alternators,  while  the  commoner  definition  involving  the  num- 
ber of  poles,  given  in  the  preceding  paragraph,  only  applies  to 
alternators  possessing  field  magnets  with  adjacent  poles  (in  the 
direction  of  rotation)  of  opposite  signs. 

The  angular  distance  of  movement  of  the  conductor  required 
to  generate  one  complete  cycle  is  called  360  Electrical  degrees. 
The  electrical  degrees  correspond  with  the  mechanical  degrees 
of  armature  rotation  only  in  a bipolar  machine  or  its  equiva- 
lent. Three  hundred  and  sixty  electrical  degrees  are  em- 
braced in  the  span  from  a given  point  in  the  magnetic  field 
under  one  pole  to  the  corresponding  point  in  the  magnetic 
field  under  the  next  pole  of  the  same  sign ; so  that  there  are  as 
many  times  360  electrical  degrees  measured  around  an  arma- 
ture as  there  are  poles  of  like  sign  in  the  related  field  magnet. 
In  speaking  of  the  movement  of  the  armature  conductor  in  the 
magnetic  field,  the  relative  movement  of  the  armature  with 
respect  to  the  field  is  here  referred  to,  and  it  may  be  obtained 


Fig.  13.  — Diagram  of  Multipolar 
Generator. 


22 


ALTERNATING  CURRENTS 


by  the  actual  rotation  of  either  the  armature  or  the  field  mag- 
net. In  modern  machines  the  latter  is  most  frequently  rotated. 

Prob.  1.  What  is  the  frequency  of  an  alternator  having  10 
pairs  of  poles  and  a speed  of  600  revolutions  per  minute?  How 
many  alternations  does  it  give  per  minute? 

Prob.  2.  An  alternator  gives  a frequency  of  25  periods  per 
second  and  lias  5 pairs  of  poles.  At  what  speed  (in  revolutions 
per  minute)  does  it  run? 

Prob.  3.  An  alternator  is  to  give  a frequency  of  60  periods 
per  second  at  a speed  of  1200  revolutions  per  minute.  How 
many  poles  must  it  have? 

Prob.  4.  Can  an  alternator  be  built  to  furnish  a frequency 
of  47  periods  per  second  when  run  at  a speed  of  600  revolu- 
tions per  minute  ? Why  ? 

Prob.  5.  Can  an  alternator  be  built  to  furnish  a frequency 
of  60  periods  per  second  when  running  at  700  revolutions  per 
minute?  Why? 


11.  Form  of  the  Voltage  Curve  of  an  Alternator.  — The  volt- 
age set  up  by  a conductor  or  coil  moving  in  a magnetic  field 
is  equal  to  the  rate  of  cutting  lines  of  force,  or  to  the  rate  of 
change  of  the  number  of  linkages  of  lines  of  force  with  the  coil, 
divided  by  108 ; or  (for  a single  conductor), 


1 deb 

io8  x dt  ’ 

where  e is  the  instantaneous  voltage  and  — is  the  time  rate  of 

dt 

cutting  lines  of  force.  In  the  case  of  an  armature  revolving  in 
a magnetic  field  as  shown  in  Fig.  12  the  voltage  will  be  at  any 
instant, 

1 S dcf> 

e — - ^ X — X — £ 


108 


P 


dt 


where  S is  the  number  of  active  conductors  on  the  armature, 
p'  is  the  number  of  paths  for  the  current  through  the  armature, 


S 

and  — is  the  number  of  active  conductors  in  series  cutting  the 

P 

lines  of  force.  If  the  armature  revolves  at  a constant  speed  of 
V revolutions  per  minute  in  a magnetic  field  of  p poles,  with 


THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


23 


useful  lines  of  force  emanating  from  each  north  pole,  the  aver- 
age voltage  set  up  may  be  deduced  from  this  formula  to  be, 


E„ 


pS®V 
p'  x 108  x 60’ 


since  is  the  average  rate  of  cutting  lines  of  force  by  each 
conductor. 

This  is  the  formula  used  to  state  the  continuous  voltage  of 
an  ordinary  direct-current  dynamo.  Since  the  voltage  produced 
by  direct-current  dynamos  is  essentially  constant,  the  effective 
and  average  values  are  equal  to  each  other  in  this  case.* 

In  the  alternating-current  dynamos  the  same  averaging  does 
not  occur  as  in  direct-current  commutating  machines  with  arma- 


ture conductors  uniformly  distributed  over  the  surface  of  the 
armature,  and  the  voltage  between  alternator  armature  ter- 
minals varies  from  instant  to  instant. 

Thus,  suppose  the  coil  of  Fig.  12  has  negligible  width  but 
comprises  S conductors  arranged  to  rotate  in  the  uniform  mag- 
netic field  as  illustrated ; then  the  voltage  in  the  conductors  is, 
at  any  instant,  as  before  stated, 


1 S dd> 
e io  sXp/Xdt' 

If  is  the  number  of  useful  lines  of  force  emanating  from  the 

north  pole,  then  under  the  conditions  described,  becomes, 
dt 

for  the  bipolar  machine, 


d(f>  _ 2 7i -r  <t> 

27 


x sin  « = —<!>  sin  a, 

pv 


where  a is  the  angular  displacement  of  the  conductor  from  a 
diameter  perpendicular  to  the  lines  of  force,  r is  the  radius  of 
the  armature,  and  T'  is  the  time  in  seconds  of  one  revolution. 

If  there  are  more  than  two  poles,  each  having  <3?  lines  of  force 
uniformly  distributed  over  its  face  so  as  to  make  a multipolar 
magnetic  field  analogous  to  the  bipolar  magnetic  field  illus- 
trated in  Fig.  12,  the  voltage  induced  is  obviously  increased  in 
proportion  to  the  number  of  poles,  and  the  last  formula  be- 
comes, in  general, 


* Jackson’s  Electromagnetism  and  the  Construction  of  Dynamos , Chap.  4. 


24 


ALTERNATING  CURRENTS 


dd)  p 2 Trr  <b 
dt  2 T 2 r 


sm  a. 


It  must  be  remembered  that  « is  an  angle  measured  with  re- 
spect to  a normal  to  the  lines  of  force,  and  it  changes  by  the 
extent  of  360°  in  passing  from  a given  point  in  the  magnetic 
field  to  the  corresponding  point  under  the  next  pole  of  like  sign. 

The  voltage  at  any  instant  now  becomes 


7T  S p 

e = — XVX72X  rpi  8111  «• 


108 
V 


p 


But  is  equal  to  where  V stands  for  the  speed  of  the 

armature  in  revolutions  per  minute.  The  formula,  therefore, 
reduces  to 

irpSQV 

e ~ 2 p'  108  x 60  Sm 

which  is  the  equation  for  a sinusoidal  or  harmonic  curve  * such 
as  is  shown  in  Fig.  1.  The  equation  may  be  written 

e = em  sin  a, 

in  which  the  maximum  ordinate,  em,  is  equal  to 

^ _ irpS^  V 

2 x p'  x 108  x 60’ 

That  is,  under  the  conditions  assumed  for  the  magnetic  field, 
and  the  arrangement  of  the  armature  conductors,  the  alternat- 
ing voltage  of  the  armature  partakes  of  the  form  of  a sine  curve. 

The  symbol  p in  this  formula  has  been  defined  as  the  number 
of  poles  comprised  within  the  magnetic  field.  Since  one  com- 
plete cycle  of  the  voltage  is  generated  while  an  armature  con- 
ductor moves  from  a given  point  under  a pole  of  one  polarity 
to  a like  point  under  the  next  pole  of  the  same  polarity,  — that 
is,  while  the  conductor  moves  through  twice  the  polar  pitch,  — 
it  is  plain  that  p in  the  last  formulas  may  be  replaced  by 

2 — /,  where/ represents  the  frequency,  and  the  formulas  become 

7T/SW  . 

« = — — TbS  sin 

p'  x 108 


and 


7 rS&f 
e™~  p'  x 108’ 


* Art.  5. 


THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


25 


The  effective  voltage  for  this  case  is 
E-  rS*-f  - 


2 22 


s*f 


V2 p'  x 108  p'  x 108 

Now,  if  it  is  assumed  that  the  field  strength  under  the  pole 
faces  illustrated  in  Fig.  12  is  altered  so  that  it  is  weakest  near 
the  center,  then  the 
rate  of  cutting  lines 
no  longer  varies  as  a 
sine  curve,  but  the 
curve  rises  more  rap- 
idly in  its  lower  por- 
tions, and  does  not 
reach  so  great  a maxi- 
mum. On  the  other 
hand,  if  the  field  is 
concentrated  towards 


AVERAGE  E 
EFFECTIVE  E 


Fig.  14.  — Flat-topped  Voltage  Curve. 


the  center,  the  curve  becomes  higher  at  the  center,  but  does 
not  rise  so  rapidly  in  its  lower  portions.  Figure  14  shows  the 

curve  of  voltage  de- 
veloped in  a coil 
which  revolves  in  a 
field  of  the  same  total 
number  of  lines  of 
force  as  that  of  Fig. 
12,  but  in  which  the 
magnetic  density  at 
the  centers  of  the 
poles  is  about  35  per 
cent  less  than  the 
average.  Figure  15 
shows  a similar  curve 
when  the  magnetic 
density  at  the  center 
is  50  per  cent  greater 
than  the  average.  In 
each  case,  the  polar 
density  is  assumed  to  change  gradually  and  uniformly  from  the 
center  to  the  edges.  In  all  cases  where  a certain  total  number 
of  lines  are  cut  per  revolution  by  a coil  revolving  at  constant 


Fig.  15.  — Peaked  Voltage  Curve. 


26 


ALTERNATING  CURRENTS 


1ig.  1(1. — Imaginary  Magnetic  Field  for  obtain- 
ing Rectangular  Voltage  Curve. 


speed,  the  average  voltage  remains  constant  regardless  of  the 
magnetic  distribution,  but  the  effective  voltage  (VAv.f2)  is  by 

no  means  independent  of 
the  distribution.  Taking 
the  maximum  voltage  of 
the  sine  curve  shown  in 
Fig.  1 as  the  unit,  the 
average  voltage*  in  each 
2 

figure  is  ~ = -637.  The 

effective  voltage  of  the 
sine  curve  shown  in  Fig.  1 

is  — - = .707.  The  maxi- 

mum  voltages  shown  in 
the  curves  of  Figs.  14  and  15  are  .84  and  1.5,  and  the  effec- 
tive voltages  are  .58  and  .84,  respectively.  If  the  field  were 
distributed  as  in  Fig.  16 
(the  total  magnetization 
remaining  constant,  and 
the  coil  of  inappreciable 
width),  the  maximum,  av- 
erage, and  effective  volt- 
ages would  be  equal  to 
each  other  (Fig.  17),  and 
of  a numerical  value  of 
.637.  This  form  of  curve 
gives  the  least  possible 
relative  effective  voltage. 




— / / / 

— 1 / / 

|(  w) I 

\ \ / / — > 

\ N.  s'  /- — > 

X.  ^ — > 

Fig.  18. 


-Magnetic  Field  for  obtaining  a Peaked 
Voltage  Curve. 


Fig.  17.  — Ideal  Rectangular  Voltage  Curve. 

On  the  other  hand,  if  the  field  is 
greatly  concentrated 
towards  the  center, 
as  in  Fig.  18,  the 
maximum  voltage  is 
very  great,  and  the 
effective  voltage  is 
considerably  greater 
than  the  average, 
though  it  by  no 
means  approaches  in 
value  the  maximum 


* Art.  5. 


THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


27 


voltage.  In  the  case  of  modem  alternators  the  windings  are 
usually  Distributed  * over  a large  portion  of  the  polar  area,  so 
that  the  resulting  curves  of  voltage  tend  to  approximate  the 
sine  form. 

12.  Equations  of  Curves  by  Fourier's  Series.  — It  has  already 
been  explained  that  the  curves  of  current  and  voltage  in  com- 
mercial alternating-current  circuits  may  deviate  widely  from 
the  sine  form,  hut  are  always  single- valued.  A general  formula 
is  needed,  therefore,  that  will  express  such  curves  whatever 
may  be  their  forms.  The  most  usual  and  satisfactory  method 
of  obtaining  such  a formula  is  by  means  of  Fourier’s  Series. 
This  series  is  based  upon  the  proposition  that  any  single- 
valued recurrent  function  may  be  represented  by  a constant 
term,  plus  a sine  term  and  a cosine  term  having  the  same  period 
as  the  function  in  question,  plus  a sine  term  and  a cosine  term 
of  one  half  the  period,  plus  a sine  term  and  a cosine  term 
of  one  third  the  period,  and  so  on  up  to  a sine  term  and  a 
cosine  term  of  one  nth  the  period  f ; where  n is  dependent  upon 
the  irregularity  of  the  curve  in  question  and  the  accuracy  of 
representation  desired. 

If/(£)  is  a single-valued  recurrent  function  of  time  having 
a period  T , it  may  be  expressed  in  accord  with  the  above  state- 
ment as  follows  : 


r a , a -27 rt  . t>  2 irt  , | . 4 7rt  . t»  4 irt 

f (t)  = A + A1  sin  — + Bx  cos  — + A2  Bin  — + B2  cos  — 


. i • 6 irt  . D 6 irt 

+ ^8  sm  ~jT  + ^3  cos  -yT 


, . ■ 2 m rt  . r,  2 nirt 

• + Aa  sin  — — b Bn  cos 


T 


T 


where  A,  Av  Bv  Av  Bv  etc.,  are  constants. 

2 irt 

Since  the  expression  represents  a proportion  of  2 7r  (or 


360°  in  angular  measure),  the  equation  may  be  written  as 
follows  : 


/(«)  = A + Al  sin  a + B1  cos  « + A2  sin  2 a +B2  cos  2 a • • • 

+ An  sin  na  + Bn  cos  na. 

The  measure  of  « at  the  end  of  a cycle  is  2 7 r. 

Evidently  /(«)  is  represented  by  a summation  comprising  a 
constant  term  plus  a series  of  sine  and  cosine  terms  having 


* Art.  20. 

t Byerly,  Fourier's  Series  and  Spherical  Harmonics , pp.  30  etseq. 


28 


ALTERNATING  CURRENTS 


maximum  instantaneous  values  of  Av  Bv  A2,  Bv  etc.,  with 
frequencies  of  one,  two,  three,  etc.,  times  that  of  the  original 
function,  and  with  the  zero  ordinate  of  each  cosine  term  oc- 
curring simultaneously  with  the  maximum  ordinate  of  the  sine 
term  of  equal  frequency.  The  value  of  the  ordinate  of  the 
periodic  curve  in  question  at  any  time,  £,  or  angle,  «,  is  equal 
to  the  algebraic  sum  of  the  ordinates  of  all  the  terms  at  the 
particular  instant.  Any  single-valued  recurrent  curve,  no 
matter  how  irregular,  may  be  thus  resolved  into  component 
sinusoids. 

a.  Examples  of  Irregular  Waves. — Figure  19  shows  a curve, 
S,  of  alternating  voltage,  which  is  composed  of  two  sine  waves 

having  respective  maxi- 
mum values  Ax  and  A3. 
The  sinusoidal  component 
of  the  same  frequency  as 
the  wave  is  called  the 
Primary  or  Fundamental 
harmonic ; remaining 
component  waves  may  in 
general  be  called  Minor 
harmonics,  or  specifically, 
the  Second,  Third.  Fourth, 
etc.,  harmonics,  depending  upon  their  frequencies  when  com- 
pared with  that  of  the  primary  wave.  In  Fig.  19  the  primary 
and  third  harmonics  only  are  present,  so  that  the  Fourier's 
Series  for  the  curve  is 


Fig.  19.  ■ 


-Periodic  Curve  composed  of  Funda- 
mental and  Third  Harmonics. 


/(«)  = y = Ax  sin  a + (—  A3 ) sin  3 a, 

where  y is  the  ordinate  of  the  curve  S for  any  time,  t , or 
angle,  a. 

The  constant  A is  zero  in  this  as  in  all  periodic  curves  having 
equivalent  loops  above  and  below  the  x axis.*  All  the  other 
terms  except  those  given  are  also  zero  as  the  sum  of  the  first 
and  third  sine  harmonics  exactly  satisfy  the  curve  S.  The 
constant  term  A3  is  negative  because  the  negative  loop  of  the 
third  harmonic  starts  at  zero  of  time.  Figure  20  shows  the  curve 
that  is  formed  by  the  same  harmonics,  but  with  the  third  har- 
monic moved  forward  180°,  so  that  A3  is  positive.  Figure  21 
* For  proof  see  Art.  13. 


THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


29 


shows  a curve  composed  of  a primary  and  a fifth  harmonic,  with 
Al  and  Ah  both  positive;  and  Fig.  22  shows  a curve  derived 
from  the  following  series  : 

x = Ax  sin  a + A3  sin  3 a -f  A5  sin  5 a. 

It  is  to  be  observed  that  the  loops  of  the  curves  shown  are 
symmetrical  on  each  side  of  a vertical  axis  drawn  through  the 


Fig.  20.  — Periodic  Curve  containing  Fundamental  and  Third  Harmonics. 

axis  of  abscissas  at  90°  or  270°.  If,  now,  a cosine  term  is  added 
to  the  components  of  Fig.  19,  for  instance,  as  in  the  expression 

y = Al  sin  a — A3  sin  3 a — B3  cos  3 «, 

which  is  shown  by  the  curves  in  Fig.  23,  this  symmetry  of  the 
loops  with  respect  to  the  vertical  axis  is  destroyed.  The  loops 


Fig.  21. — Periodic  Curve  containing  Fundamental  and  Fifth  Harmonics. 


of  this  type,  as  of  the  preceding  curves,  are,  however,  sym- 
metrical one  with  the  other.  That  is,  if  the  negative  loops  are 
revolved  180°  on  the  axis  of  abscissas  so  as  to  stand  between 
the  positive  loops,  all  of  the  loops  look  just  alike,  and  if  one 


80 


ALTERNATING  CURRENTS 


of  the  negative  loops  is  then  moved  along  the  axis  one  half  a 
period  so  as  to  lie  upon  one  of  the  positive  loops,  it  will  exactly 
coincide  with  a positive  loop.  From  the  brief  description 
already  given  of  the  arrangement  of  armature  windings  and  uf 
the  magnetic  field  of  the  ordinary  commercial  alternator,*  it  is 


Fig.  22. — Periodic  Curve  containing  Fundamental,  Third,  and  f ifth  Harmonics. 

seen  that  the  same  magnetic  lines  are  cut  in  the  same  order  by 
exactly  the  same  arrangement  of  conductors  during  the  gener- 
ation of  both  the  positive  and  negative  loops  of  voltage  which  are 
as  a result  alike  in  area  and  form,  provided  the  machine  is 


Fig.  23.  — Periodic  Curve  containing  Sine  and  Cosine  Terms  in  the  Third 
Harmonic. 

mechanically  and  magnetically  rigid.  It  will  likewise  be 
shown  later  that  the  negative  and  positive  current  loops  also 
ordinarily  have  the  same  characteristic  likeness.  It  is  there- 
fore evident  that  such  waves  can  be  expressed  by  such  formulas 
as  have  preceded  and  which  include  only  the  odd  harmonics. 


* Arts.  10  and  11. 


THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


31 


If  the  even  harmonics  appear,  this  characteristic  symmetry 
no  longer  exists.  Figure  24  shows  a curve  answering  to  the 
formula : 

y = sin  a + A2  sin  2 a. 


Fig.  24. — Periodic  Curve  Containing  Fundamental  and  Second  Harmonics. 

Figure  25  shows  a curve  compounded  of  the  first,  secondhand 
third  sine  harmonics  ; and  Fig.  26  has  first  and  second  sine 
and  second  cosine  harmonics  — the  latter  two  combined  into 


Fig.  25.  — Periodic  Curve  Containing  Fundamental,  Second,  and  Third  Harmonics. 

one  curve,  which  may  be  done  by  adding  their  ordinates 
algebraically.  The  resultant  irregular  curves  in  these  figures 
do  not  have  positive  and  negative  loops  which  will  superpose 
by  revolution  around  and  sliding  along  the  axis  of  abscissas  as 


32 


ALTERNATING  CURRENTS 


above  described,  because  for  each  half  period  of  the  primary 
curve  the  even  harmonics  complete  one  or  more  entire  cycles 
and  have  relatively  opposite  effects  upon  the  positive  and  nega- 


Fig.  26.  — Periodic  Curve  having  Second  Harmonic  which  is  composed  of  Sine 
and  Cosine  Terms. 

tive  loops  of  the  primary  curve.  On  the  other  hand,  the  odd 
harmonics  keep  at  all  times  the  same  relation  to  the  primary, 
i.e.  when  the  primary  reverses,  the  odd  harmonics  also  reverse. 

Prob.  1.  Construct  the  voltage  curve  expressed  by  the 
formula  e = 100  sin  « + 50  cos  a + 30  sin  3 a + 20  cos  3 a. 

Prob.  2.  Construct  the  curve  of  current  expressed  by  the 
formula  i = 200  sin  a — 25  sin  5 « + 15  cos  5 «. 

Prob.  3.  Construct  the  voltage  curve  expressed  by  the 
formula  e = 200  sin  a + 200  cos  a + 50  sin  3 a + 50  cos  3 « 

---  25  sin  7 « — 25  cos  7 a. 

Prob.  4.  Construct  the  curve  represented  by  the  formula 
y = 50  + 50  sin  a + 40  cos  « + 30  sin  2 a + 20  cos  2 a. 

h.  Fourier's  Series  with  Sine  and  Cosine  Terms  Combined. — 
Sine  and  cosine  harmonics  of  equal  frequency  may  be  combined 
into  one  curve,  as  explained  with  reference  to  Fig.  26,  and  a 
sine  term  is  formed  thereby  having  a maximum  which  may  be 
designated  C and  which  is  out  of  phase  with  the  original  sine 
component  harmonic  by  some  angle  /3„.  As  a result  Fourier's 
Series  may  be  expressed  in  a more  general  form,  thus: 

f = A.  -f-  C±  sin  + /3\)  ~b  sin  (2  a.  + fdf)  ■ • • 

+ Cn  sin  Qua  + /3„)  ; 


THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


33 


or  where  /3,[  is  the  angle  between  the  new  harmonic  and  the 
original  cosine  component, 

/(«)  = A + <7j  cos  («  — ySj)  + C2  cos  (2  « — /Sg)  ••• 

+ On  cos  (na  - /3') . 

In  these  expressions  On  = A\  + B\,  tan  /3re  = and 

tan  /3'„=  —5,  since  (7„  is  the  resultant  of  An  and  Bn,  which  differ 
B» 

from  each  other  in  phase  by  90°.  It  is  usual  in  drawing  the 
harmonics  of  a curve  to  plot  the  total  harmonics  as  given  by 
either  of  these  formulas  rather  than  to  use  their  sine  and  cosine 
components,  which  add  to  the  complication.* 

13.  Determination  of  the  Value  of  the  Constants  in  Fourier’s 
Series.  — The  value  of  Fourier’s  Series  to  electrical  engineer- 
ing consists  in  the  possibility  of  resolving  irregular  waves  of 
current  and  voltage  into  their  harmonics  and  by  that  means 
obtaining'  mathematical  expressions  to  which  the  elementary 
laws  of  electricity  can  be  applied.  To  do  this  it  is  evidently 
necessary  to  obtain  the  numerical  values  of  the  constants  con- 
tained in  the  formulas. 

Having  given  the  formula, 

/(a)  = A + Ax  sin  a + Bx  cos  a + A2  sin  2a  + 52  cos  2 a ••• 

+ An  sin  na  + Bn  cos  na , 
an  expression  for  the  value  of  A may  be  obtained  by  multiply- 
ing both  sides  of  the  equation  by  da , and  integrating  between 
0 and  2 7 r.  Thus  : 


X%r  r /~2rr 

f(a)da  = Aj  da-\-AxJ  sin  ada  -f  Bl  cos  ada--- 


+ A 


0 si 


sin  tiada  + B 


l ( 


cos  nada. 


* A theoretical  discussion  of  Fourier’s  Series  will  be  found  in  the  first  three 
chapters  of  Byerly,  Fourier's  Series  and  Spherical  Harmonics ; J.  W.  Mellor, 
Higher  Mathematics , Chap.  VIII ; Merriman  and  Woodward,  Higher  Mathe- 
matics ; and  similar  works.  The  following  articles  are  of  considerable  interest 
in  this  connection:  Periodic  Functions  Developed  in  Fourier  Series;  The 
Graphical  Method,  by  Professor  John  Perry,  London  Electrician , Vol.  35,  p.  285  ; 
Wave  Form  Synthesis,  London  Electrician,  Vol.  35,  p.  257  ; Trans.  Amer.  Inst. 
Elect.  Eng.,  Vol.  12,  p.  476  ; Fleming’s  Alternate  Current  Transformer  in 
Theory  and  Practice,  Vol.  II,  p.  454  ; Graphical  Analysis  of  Harmonic  Curves, 
by  Wedmore,  London  Electrician,  Vol.  35,  p.  512 ; Approximate  Method, 
Houston  and  Kennedy,  Electrical  World,  Vol.  16,  p.  580  ; Graphical  Compu- 
tations in  Alternating  Current  Circuits,  Eclairage  Electrique,  Vol.  16,  p.  397. 


34 


ALTERNATING  CURRENTS 


All  the  terms  on  the  right,  except  the  first,  reduce  to  zero,  as 
may  be  shown  by  integrating  the  general  sine  and  cosine  terms 
as  follows  : 


A 


Jo  81 


sin  nada  — Ar, 


1 V* 

- cos  na  = 0. 
n Jo 


Therefore 


J--277  f \ \2n 

cos  nudu  — An  [ + - sin  na)  =0. 
u V n Jo 

A=jzr^d'1- 


en- 


Jr*  2n 

/(«)  da  is  the  net  area  of  the  curve  for  an 

0 

tire  period,  and  2 it  is  its  base ; so  that  A is  the  average  ordi- 
nate for  an  entire  period.  Since,  as  has  already  been  explained, 
the  positive  and  negative  loops  in  the  ordinary  alternating  elec- 

i n* 

trie  voltage  and  current  curves  are  closely  alike,  - — I /(«)  da 

2 IT 

reduces  to  zero  because  the  area  of  the  positive  loop  in  a period 
is  equal  to  the  area  of  the  negative  loop.  The  constant  A , 
therefore,  need  not  be  included  in  writing  the  series  for  compu- 
tations of  such  quantities ; that  is,  the  axis  of  abscissas  is 
placed  symmetrically  with  respect  to  the  positive  and  negative 
areas. 

To  obtain  Ax  multiply  the  series  by  sin  ada  and  integrate 


between  the  same  limits  as  before.  Thus 


sin  ada 


J^2n 

, /(«)si 

X2n  f2n  _ ^ /*2ir 

s\n  ada  + Ax  I sin2  ada  + BXJ  cos  a sin  ada 

J'2"  C*2tt 

# sin  na  sin  ada  + Bn  J cos  na  sin  ada. 

Each  of  the  terms  in  the  right-hand  member  of  the  equation 
except  the  second  term  is  of  the  form  of  one  of  the  follow- 
ing : 

Jr2n  _ P*2tt 

I sin  ada , I sin  a cos  ada, 
o 

X2  7T  _ _ /'2^- 

sin  pa  sin  qada,  J sin  pa  cos  qada, 

/*2  7T  .^2ir 

I sill  pa  cos pada,  or  | cos  pa  cos  qada. 


where  p and  q are  unequal  integers.  As  is  well  known,  these 
expressions  reduce  to  zero.  This  is  equivalent  to  saying  that 


THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


35 


any  curve  which  may  be  represented  by  one  of  these  expres- 
sions incloses  as  much  negative  area  below  the  axis  of  abscissas 
as  positive  area  above  the  axis,  in  any  length  equal  to  one 
period  measured  along  that  axis.  The  rule  also  applies  for  full 
periods,  when,  of  p and  q , one  is  an  integer  and  the  other  dif- 
fers from  an  integral  number  by  one  half. 

The  second  term  in  this  equation  is  of  the  form 


X2tt 

sin2^«d«, 


where  p is  any  integer.  This  reduces  to  i r,  as  also  does  the 
analogous  expression 


J^2n 
0 ( 


J"2ir 
0 * 


cos  2pada. 

The  original  equation  therefore  reduces  to 
f (a)  sin  ada  = ttAv 

1 r2n 

Ax  = - I f (a)  sin  ada. 

TT 

A like  process  in  which  the  original  equation  is  multiplied  by 
cos  ada , and  reduced  to  find  the  expression  for  Bl  by  sin  2 ada, 
and  reduced  to  find  the  expression  for  Av  etc.,  shows  that  the 
expressions  representing  the  various  constants  are  as  follows : 


and 


1 r 2*- 

B,  — — I /(«)  cos  ada, 

7W0 

1 C2k 

A2  = — /(a)  sin  2 ada, 

Bn  = — f f (a)  cos  2 ada, 

7T»/o 


An=  - I /(a)  sin  nada, 

IT  i/O 

1 r2w 

I /(«)  cos  nada. 

7T»/0 

14.  Method  of  Finding  the  Harmonics  of  a Periodic  Curve  by 
Fourier’s  Series.  — If  f («)  (that  is,  each  ordinate  of  the  re- 
current curve)  is  dependent  upon  a by  some  known  law,  its 
value  can  usually  be  incorporated  into  the  formulas  for  the 
constants  given  above  and  the  integration  performed.  For 


36 


ALTERNATING  CURRENTS 


instance,  such  would  be  the  case  if  the  loops  of  the  curve  were 
rectangular.  In  the  case  of  current  or  voltage  loops  such  a 
relation  between  the  ordinates  and  abscissas  of  the  curve  is  not 
apt  to  exist,  so  that  it  is  necessary  to  change  the  integral  sign 
into  a summation  sign  and  obtain  the  result  as  accurately  as 
time  will  permit  or  necessity  require. 

Tims  erect  m equally  spaced  ordinates  on  the  axis  of  abscissas 
of  an  experimentally  determined  curve,  beginning  at  the  incep- 
tion of  a period.  The  ordinates  divide  the  base  line  into  m — 1 
equal  divisions  in  the  period.  The  lengths  of  the  ordinates, 
from  the  axis  of  the  abscissas  to  the  curve,  may  be  called  e0,  ev 
•••  em.  The  base  of  one  complete  period  being  2 v radians  in 


Fig.  27.  — Showing  Periodic  Wave  divided  into  Elementary  Areas  for  obtaining 

Harmonics. 


length,  the  distance  between  any  adjacent  two  of  the  equally 

2 t r . 

spaced  ordinates  is  -radians  (see  Fig.  27).  Then  the  value 

m — 1 

1 C2,t 

A,  = — I /(«)  sin  ada  can  be  written  approximately 

1 7 rJn' 


2 

1 m — 1 Z— ■ -i 


e,.sin  [ k 


vi 


where  ek  represents  the  ordinate  of  the  curve  or  ,/(«)  at  any 
particular  division  point,  k the  designating  number  of  the  cor- 

2 7T 

responding  ordinate,  and  — distance  between  adjacent  ordi- 

, vi  — 1 

nates. 

Hence, 

9 

A = -Ar 

m — 1 


e , sin 


Vi 


— 1/ 


■ + sin  2 


vi  — 1 


+ em  sin 


f 2 ’ 

m — 


V vi  — 1 /_ 


THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


where  ev  e2 , etc.,  are  the  ordinates  of  the  curve  at  the  division 
points,  1,  2,  3,  •••  m,  and  in  general 


B = 


on  — 1 
o 


m- 


■ 1 


• / 2tt  V . f0  2tt  ) , . ( 

e.sin  n- + e9sin  In — Hemsin  nm 

1 V m — lj  2 V m-lj  \ 

f 2 IT  \ f 0 2 7T  \ ( 

e.cos  n +e9cos  2 n — be»cos  : 

1 V m-lj  2 V m-lj 


\ on  — 1 


V m — 1 


The  positive  and  negative  loops  of  ordinary  alternating  cur- 
rents and  voltages  are  similar,  and  for  them  it  is  only  necessary 
to  divide  up  one  loop ; and  for  the  same  reason  even  harmon- 
ics of  the  series  are  commonly  unimportant.  The  larger  m 
is  made,  the  more  accurate  will  be  the  result ; likewise  the  ac- 
curacy is  increased  by  making  n large,  though  it  is  seldom  nec- 
essary to  obtain  more  than  the  first,  third,  and  fifth  harmonic. 

Having  found  Av  A3,  etc.,  and  Bv  B3,  etc.,  the  values  of  Cv 
C3,  and  /3V  fi3  may  be  at  once  obtained  for  substitution  into  the 
general  formula.*  Having  obtained  these  values,  the  harmonics 
represented  by  each  term  of  the  series  can,  if  desired,  be  plotted 
with  the  periodic  curve  to  which  it  belongs.  It  must  be  re- 
membered in  doing  this  that  a = 0 when  the  irregular  curve 
crosses  the  X axis  in  an  upward  or  positive  direction. 

The  values  of  the  constants  may  be  obtained  approximately 
by  the  use  of  a planimeter  instead  of  by  the  laborious  com- 
putation involved  in  solving  the  foregoing  equations.  Meas- 
ure the  ordinates  ev  e2.  e3,  •■■em  as  before  and  multiply  each 
by  the  sine  of  the  proper  angle  as  shown  in  the  formula  of 
the  desired  constant  or  Av  or  Bx  or  B2 , etc.).  Plot  a 
curve  over  the  points  1,  2,  3,  4,  ■■■m  with  ordinates  equal  to 

( 2 

these  products,  that  is,  place  the  ordinate  ex  sin  [ n 


m — 1 


m — 1 


, over  the  point  1,  e2  sin  (2  n 


m — 1 


or 


e9  cos  Z n 


on  — 1 


, over  the  point  2,  and  so  on  (see  curve  z in 


Fig.  28).  Measure  the  areas  of  the  resultant  loops  by  means 
of  a planimeter  and  add  them  algebraically,  considering  areas 
above  the  axis  of  abscissas  as  positive  and  those  below  the  axis 
as  negative.  If  m is  sufficiently  large,  this  area,  lT,  will  be 
with  sufficient  accuracy 


* Art.  12  6. 


38 


ALTE R NATING  C UR RENTS 


/'2n 

U = I /(«)  sin  nudu  or  ( f (a)  cos  nudu. 

«/0  %)  0 

i rn  . . u 

Hence,  An=—  I /(a)  sin  nudu.  = — . 

7T  *Z°  7T 

In  Fig.  28  is  shown  a recurrent  curve,  X , and  the  process 
of  obtaining  H3  is  indicated.  To  do  this,  37  values  of  e and 
the  corresponding  37  values  of  sin  3 u were  used,  u being  re- 

2 77"  2 7T  i • 

spectively  0,  — r,  2 — , etc.,  at  the  successive  points.  These 
36  36 

values  multiplied  into  values  of  e,  ev  ev  etc.,  and  plotted  on  the 
corresponding  ordinates  give  the  points  on  the  curve  Z.  If 
the  areas  of  the  positive  and  negative  loops  of  curve  Z are  meas- 


Fig.  28.  — Product  of  an  Alternating-current  Curve  with  Sine  Third  Harmonic 
Analyzer,  x is  the  Irregular  Curve,  and  z the  Curve  of  Products. 

ured  by  a planimeter,  it  will  be  found  that  there  is  a small  posi- 
tive area  in  excess.  The  constant  A3  is  therefore  positive : i.e. 
the  sine  component  of  the  third  harmonic  starts  at  zero  with  a 
positive  loop,  and  has  a maximum  height  equal  to  twice  the 
length  obtained  by  dividing  the  algebraic  sum  of  the  areas  of  the 
loops  of  curve  Z by  the  length  of  the  base  for  the  period.  This 
is  readily  converted  into  volts  by  applying  the  scale  used  in 
plotting  the  values  of  e.  The  dotted  line  in  Fig.  28  shows  the 
harmonic  plotted  in  position  and  to  scale. 

The  other  constants  may  be  obtained  by  following  a similar 
process.  While  the  foregoing  example  is  applied  to  a com- 
plete period  of  the  curve,  it  is  sufficient  to  carry  out  the  analy- 
sis on  a single  loop  or  half  period  of  ordinary  alternating 
current  or  voltage  waves  which  have  half  periods  alike,  as 
already  pointed  out. 


THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


39 


14  a.  Components  of  Alternating  Curves  do  not  always  fall 
within  the  Fourier  Series. — The  foregoing  articles  deal  with 
the  harmonic  analysis  of  alternating  voltage  and  current  curves 
on  the  hypothesis  that  the  components  of  such  curves  are  all  of 
frequencies  represented  by  integral  numbers  of  times  the  pri- 
mary frequency,  and  this  is  an  accurate  hypothesis  with  respect 
to  curves  produced  under  perfectly  stable  recurrent  conditions, 
such  as  voltage  curves  induced  by  the  uniform  rotation  of  the 
armature  of  a synchronous  generator  constructed  with  perfect 
mechanical  rigidity  and  possessing  a perfectly  rigid  magnetic 
field.  But  these  conditions  do  not  always  exist  and  sometimes 
are  departed  from  widely.  The  rotation  velocity  may  lack 
uniformity,  the  shaft  of  the  armature  may  spring  and  alter  the 
magnetic  conditions,  and  other  occurrences  may  likewise  intro- 
duce influences  into  the  voltage  curve  that  are  independent  of 
the  primary  frequency.  Also,  even  though  the  voltage  wave 
should  be  free  from  such  effects,  variations  in  the  circuit 
through  which  the  current  flows  may  introduce  distortions  into 
the  current  wave  that  are  independent  of  the  primary  frequency. 

Variations  of  the  kind  here  referred  to  are  generally  not 
strictly  recurrent,  but  this  may  not  be  indicated  by  the  analysis 
of  a single  period  of  the  curve  and  is  not  likely  to  be  indicated 
by  the  analysis  of  a half  period.  As  such  variations  are  likely 
to  cause  the  successive  loops  to  differ  in  form  and  area,  the 
analysis  of  several  periods  is  likely  to  indicate  their  existence 
and  magnitude.  They  are  generally  too  small  a factor  in  most 
of  the  existing  commercial  conditions  to  make  them  of  serious 
import,  but  in  case  of  necessity  they  may  be  treated  like  inter- 
ference phenomena  which  introduce  recurrent  effects  embracing 
several  or  many  of  the  periods  of  the  alternating  curve  in  ques- 
tion and  have  a frequency  which  is  some  fractional  multiple  of 
the  primary  frequency. 

15.  Concrete  Examples  of  the  Resolution  of  an  Alternating- 
current  Curve  into  its  Harmonic  Components.  — The  positive 
loop  of  a voltage  wave  of  an  alternator  is  represented  by  the 
heavy  line  of  Fig.  29,  and  the  constants  of  Fourier’s  Series  for 
this  curve  have  been  determined  up  to  the  seventh  harmonic, 
giving  the  values, 

A1  = + 98.6, 

Aa=-  13.3, 


A = -14.7, 
Bz  = 8-18.2, 


40 


ALTERNATING  CURRENTS 


A = ~ !-6. 

A7  = + 0.56. 


B5  = - 4.8, 
B7  = + 1.2. 


The  equation  for  the  curve  as  determined  by  this  means  is 

e =98.6  sin  « — 13.3  sin  3 « — 1.6  sin  5 a + 0.56  sin  7 « 

— 14.7  cos  a + 18.2  cos  3 « — 4.8  cos  5 a + 1.2  cos  7 a. 

Substituting  various  values  of  a in  this  equation,  the  corre- 
sponding values  of  e are  given,  and  the  corresponding  curve, 
which  is  dotted,  has  been  plotted  in  the  figure.  It  will  be 
noticed  that  the  calculated  curve  very  closely  approximates  to 


0°  q e2  q q e q c7  lso' 

Fig.  29.  Irregular  Curve  of  Voltage,  and  Calculated  Curve  composed  of  tlie  First,  Third 

Fifth,  and  Seventh  Harmonics. 


the  exact  form  of  the  original  curve.  If  a larger  number  of 
constants  had  been  determined,  the  calculated  curve  would 
have  crossed  the  original  curve  a larger  number  of  times,  and 
the  approximation  would  have  been  still  closer.  The  cal- 
culated curve  must  coincide  with  the  original  curve  at  each 
ordinate  which  was  used  in  the  computation.  The  calculated 
curve  cannot,  except  in  special  cases,  exactly  coincide  with  the 
original  curve  unless  m = ac.  The  series  used  is  rapidly  con- 
vergent, and  in  this  particular  curve  the  effect  of  the  fifth  and 
seventh  harmonics  is  quite  small,  so  that  the  curve  is  sufficiently 
well  represented  for  practical  purposes  by  the  fundamental  and 
third  harmonics,  in  which  case  the  equation  is 


THE  VOLTAGE  DEVELOPED  BY  ALTERNATOKS 


41 


e = 98.6  sin  « — 13.3  sin  3 a 
— 14. T cos  « + 18.2  cos  3 «. 

The  corresponding  sine  and  cosine  terms  of  the  series, 


A,  sin  a ) ( A„  sin  3«]  f A,  sin  5 « 

+ + 

B1  cos  « J [ + B3  cos  3 a J [ + B5  cos  5 « 


+ 


\ + etc., 
i + etc., 


may  be  conceived  as  representing  the  rectangular  sine  compo- 
nents of  the  terms  of  a single  sine  or  cosine  curve.*  The  equa- 
tion when  reduced  to  the  more  general  form  (using  /3)  has  the 
following  constants  : 

C\  = 99.7,  /3X  = - 8°  29', 

C3  = 22.5,  /S3  = - 53°  50', 

C5  = 5.1,  Bh  = + 71°  34', 

G1  = 1.3,  /37  = + 66°, 

and  the  equation  is 


e = 99.7  sin  («  - 8°  29')  - 22.5  sin  (3  a - 53°  50') 

— 5.1  sin  (5  a + 71°  34')  + 1.3  sin  (7  a + 66°), 

and  its  value  to  a considerable  degree  of  approximation  is 
e = 99.7  sin  («  - 8°  29')  - 22.5  sin  (3  « - 53°  50'). 

The  example  which  has  been  taken  fairly  represents  the 
complexity  of  the  average  distorted  waves.  Some  alternating- 
current  curves  are  so  greatly  distorted  that  larger  numbers  of 
terms  of  the  series  are  required  to  closely  represent  them,  but  for 
practical  purposes  three  or  four  terms  are  generally  sufficient. 
In  a large  number  of  waves  the  forms  are  so  simple  that  two 
terms  of  the  series  give  sufficient  approximation  for  practical 
purposes. 

Following  are  examples  of  the  calculation  of  the  constants  of 
these  equations : 

m = 9,  (-^r)  = 221°. 

\m  — 1/ 

Values  of  e from  curve  by  measurement : 

ex  = 18,  e3  = 70,  e5  — 125,  e7  - 30. 
e2  = 42,  = 110,  e6  — 80,  e0  = es  = 0. 

a _ i j 18  sin  22|°  + 42  sin  45°  + 70sin67|°  + •••!  _ qn 
1 ~ 1 1 + 30  sin  1571°  “ J “ ’ 


42 


ALTERNATING  CURRENTS 


, . f 18  sin  674°  + 42  sin  135°  + 70  sin  2021°  "I  100 

A‘  = i{  + ..  + 808inll2i»  = j = - 13.3, 

Q 1 f 18  cos  674°  + 42  cos  135°  + 70  cos  2024°  + •••  ] 1QO 
^“‘i  '+30coS112>°  ' |=18'2’ 

= v'A;?TB7-  = 22.5,  /3,=  tan  1 ^*  = - 53° 50.' 

A 

As  C is  a second  root,  it  is  algebraically  either  positive  or 
negative,  but  the  conditions  of  the  problem  require  it  to  enter 
the  equation  with  the  algebraic  sign  pertaining  to  the  corre- 
sponding A. 

Prob.  1.  Find  the  primary  and  third  harmonics  of  an  equi- 
lateral triangular  voltage  wave  having  an  altitude  of  10  volts. 

Prob.  2.  Find  the  primary  and  third  harmonics  of  a voltage 
wave  which  is  in  the  form  of  a semicircle  with  a diameter  of  10. 

16.  Effective  Value  of  an  Irregular  Curve  of  Voltage  or  Cur- 
rent. — The  effective  ordinate  of  an  alternating-current  curve 
may  be  determined  by  integrating  directly  from  its  equation. 
The  effective  value  squared  is 

^2=Av.e2=-  CeHa 
7 r*4o 

_ 1 C,r(A1  sin  « + Az  sin  3 « + etc.  + _Bj  cos  « + B3  cos  3 « 

— 77-  J0  4-  etc.)2  da; 

X7T  > S*1 7 " /»7T 

sin  pa  sin  qada,  sin^a  cos  qada,  | cosy>«  cos  qada, 

J f sin  px  cos  pxda  are  zero  when  p and  q are  unequal  integers, 
there  results 

1 rn  A 2 a 2 /**■ 

E 2 = — ( e2da  = — L ( sin2  ad  a + — 3-  ( sin2  3 ada 

rjp*J  0 77  0 77  0 

A. 2 Cn  B 2 Cn 

sin2  5 ada  + etc.  4 L I cos2  ada 

7T  *'0  7T  •/> 

B 2 B 2 C17 

4 3.  I cos2  3 udu  4 5-  I cos2  5 ada  4-  etc. 

7 r c/0  77  c/0 


4 2 , ,6  2 , i?„2 , 5 2 , , 

-y-  4-  etc.  4-  — 4-  — y-  4-  4-  etc., 


ButX 

7T  ( %7r 

sin2  pud  a and  1 cos2 

hence, 

A 2 1 2 

pj2  = Ai.  + th_  + 

and 

(S^K=nA2 

E={LK,^- 

THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


43 


The  effective  value  of  the  curve  is  therefore  independent  of  /S. 
The  area  and  the  mean  ordinate  of  the  curve,  on  the  other  hand, 
are  not  independent  of  /3,  which,  of  course,  depends  on  the 
relative  magnitude  and  algebraic  signs  of  the  sine  and  cosine 
functions.  That  is,  the  effective  value  of  the  current  depends 
only  on  the  amplitudes  of  the  harmonics,  and  is  not  influenced 
by  their  relative  positions ; but  the  average  value  depends 
jointly  upon  the  amplitudes  and  relative  positions  of  the  har- 
monics. 

The  following  table  gives  various  ratios  relating  to  periodic 
curves  of  various  forms  with  known  equations  or  which  are 
illustrated  in  various  figures  in  this  book : 

TABLE 


Characteristic  Features  of  Different  Forms  of  Alternating- 
current  and  Voltage  Curves 


Name  of  Curve 

Ratio  of 
Average  to 
Maximum 
Value 

Ratio  of 
Effective  to 
Maximum 
Value 

Ratio  of 
Maximum  to 
Effective 
Value 

Ratio  of 
Effective  to 
Average 
Value 

Area  of 
Squared 
Curve  corre- 
sponding to 
Unit  Maxi- 
mum Value 

Triangle 

.500 

.577 

1.732 

1.155 

.333 

Sinusoid 

.637 

.707 

1.414 

1.112 

.500 

Parabola 

.666 

.730 

1.369 

1.096 

.533 

Semicircle 

.785 

.816 

1.225 

1.040 

.667 

Rectangle 

1.000 

1.000 

1.000 

1.000 

1.000 

Prob.  1.  Find  the  effective  value  of  a voltage  having  a wave 
expressed  by  the  formula  e = 50  sin  (a  + 30°)  -f  30  sin  (3  a + 
60°)+  20  sin  (5  a + 45°). 

Prob.  2.  Find  the  effective  value  of  a current  curve  ex- 
pressed by  the  formula  i = 200  sin  (a + 90°)  — 50  sin  (3  a-f  20°). 

17.  Alternating  Current  Voltmeters  and  Amperemeters  measure 
Effective  Volts  and  Amperes.  — Alternating  voltages  and  cur- 
rents cannot  be  measured  by  instruments  depending  upon  per- 
manent magnets,  so  that  instruments  for  such  purposes  usually 
fall  within  three  general  classes  : * 

1.  Electromagnetic  instruments,  depending  on  the  magnetic 
attractions  of  two  coils,  one  fixed  and  the  other  movable, 
* Jackson’s  Elementary  Electricity  and  Magnetism , Chaps.  VIII  and  XIV. 


44 


ALTERNATING  CURRENTS 


carrying  the  current  to  be  measured ; or  of  a fixed  coil  or  coils 
and  a small  movable  vane-like  bit  of  iron  or  disk  of  conducting 
metal.  The  coils  may  or  may  not  be  provided  with  soft  iron 
cores,  with  the  iron  carefully  subdivided  to  prevent  excessive 
eddy  currents. 

2.  Hot  ivire  instruments,  in  which  the  expansion  of  a wire 
due  to  the  heating  effect  of  a current  is  used  to  obtain  a 
deflection. 

3.  Electrostatic  instruments,  depending  upon  electrostatic 
forces  for  obtaining  a deflection,  as  in  an  electrometer. 

In  the  first  two  classes  the  force  tending  to  deflect  a needle, 
suitably  attached,  is  proportional  to  the  square  of  the  current 
flowing  in  the  coil  or  wire,  and  hence  is  also  proportional  to  the 
square  of  the  voltage  at  the  terminals  of  the  instrument  when 
it  is  arranged  for  use  as  a voltmeter,  except  in  the  cases  of 
instruments  with  iron  cores  or  vanes,  in  which  the  forms  of 
pole  pieces  or  the  saturation  of  the  iron  may  be  caused  to 
modify  the  force.  In  the  third  class  the  tendency  of  the  needle 
to  deflect  is  proportional  to  the  square  of  the  voltage-  im- 
pressed upon  the  terminals  ; and  it  is  proportional  to  the  square 
of  the  current  flowing'  through  a shunt  across  the  terminals 
of  the  instrument,  if  it  is  arranged  with  a shunt  for  the  pur- 
pose of  using  it  as  an  amperemeter. 

When  an  instrument  of  either  of  these  types  is  inserted  in 
an  alternating-current  circuit,  it  tends  at  each  instant  to  deflect 
its  needle  by  a force  proportional  to  the  square  of  the  instanta- 
neous current  flowing  through  it  or  instantaneous  voltage  im- 
pressed upon  its  terminals.  The  deflection  of  the  needle  is 
resisted  by  a spring  or  like  device  affording  a resisting  force 
proportional  to  the  extent  of  movement.  On  account  of  the 
great  frequency  of  commercial  alternating-current  circuits  the 
resultant  deflection  of  the  needle  is  ordinarily  constant  and  is 
proportional  to  the  average  of  the  squared  instantaneous  values 
of  the  current  or  voltage.  The  square  root  of  this  deflection  is 
proportional  to  the  effective  amperes  or  volts  impressed  upon 
the  instrument. 

Such  an  instrument  can  therefore  be  calibrated  to  read 
effective  amperes  or  volts.  When  the  instrument  is  so  cali- 
brated, it  will  read  direct  or  alternating  currents  or  voltages 


THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


45 


on  the  same  scale,  since  the  force  causing  the  needle’s  deflection 
is  in  every  case  proportional  to  the  average  of  the  squares  of 
the  instantaneous  values  of  the  current  or  voltage.  It  is 
further  evident,  for  the  same  reason,  that  instruments  of  these 
types  will  correctly  measure  alternating  currents  or  voltages 
whatever  may  be  the  forms  of  their  waves,  provided  that  modi- 
fying influences  from  iron  cores  or  similar  extraneous  influ- 
ences are  not  allowed  to  creep  in. 

Direct  currents  and  voltages  and  effective  alternating  cur- 
rents and  voltages  are  uniformly  indicated  throughout  this 
hook  by  the  same  symbols,  I and  U,  since  they  are  equivalent 
in  their  electro-dynamic  or  heating  effects. 

18.  The  Effective  Voltage  developed  by  an  Alternator.  — In 
Art.  11  it  is  shown  that  the  instantaneous  voltage  of  a multi- 
polar armature  generating  a sine  voltage  curve  is 


irpSQ  V 

2 xp'  x 108  x 60 


sin  a = — sin  cu 

p'  x 108 


and  that  then  the  maximum  voltage  is 

7rpS<b  V _ 

2 x p'  x 108  x 60  p'  x 108 

To  obtain  the  effective  voltage  of  a machine  which  fulfills 
the  conditions  of  this  formula  it  is  only  necessary  to  divide  by 
the  square  root  of  2,  since  that  is  the  ratio  between  the  maxi- 
mum and  effective  values  of  the  sine  voltage  curve. 

Therefore, 

em  ^ 1.11  p#I>  V ^ 2.22 +SW 
V2  p'  x 108  x 60  p'  x 108 

The  numerical  coefficient  in  this  formula  is  accurate  only  for 
a machine  with  very  narrow  coils  and  with  such  a distribution 
of  magnetism  that  the  voltage  curve  is  of  the  sine  form.  It  is, 
however,  correct  within  a few  per  cent  for  a large  proportion 
of  the  alternators  of  the  present  day,  as  will  be  seen  later.* 


Prob.  1.  What  is  the  voltage  generated  by  an  alternator 
having  500  active  conductors  in  series,  a magnetic  flux  from 
each  pole  of  2,000,000  lines  of  force,  a speed  of  1200  revolu- 
tions per  minute,  and  five  pairs  of  poles,  supposing  the  formula 
above  to  be  applicable  ? 


* Art.  20. 


46 


ALTERNATING  CURRENTS 


Prob.  2.  The  magnetic  field  of  an  alternator  has  been  de- 
signed with  six  pairs  of  poles  so  that  2,000,000  lines  of  force 
emanate  from  each  pole.  The  armature  is  to  run  at  500  revo- 
lutions per  minute  and  generates  a sinusoidal  voltage.  What 
must  be  the  number  of  conductors  in  series  to  give  1000  volts? 

19.  Comparison  of  the  Voltages  developed  by  an  Alternator 
and  a Direct-current  Dynamo. — In  multipolar  direct  current 
dynamos 

E_  PS<$>V 
p'  x 108  x 60 

Assuming  two  machines  in  which  /S',  p , p' , <F,  and  V are  alike, 
one  of  which  has  the  armature  conductors  uniformly  spaced 
over  the  armature  surface  and  produces  direct  currents,  and  the 
other  of  which  has  the  conductors  arranged  in  a narrow  coil 
and  produces  alternating  currents,  the  formulas  show  that  the 
average  voltage  developed  in  the  alternator  armature  is  equal 
to  that  developed  in  the  direct-current  armature,  but  the  effec- 
tive voltage  of  the  alternator  should  be  greater  than  the  aver- 
age voltage.  Calling  the  ratio  of  effective  to  average  voltage  k, 
it  is  seen  that  the  effective  voltage  of  the  alternator  is  k times 
that  of  the  direct-current  machine  under  the  conditions  repre- 
sented by  the  formulas.  The  ratio  of  effective  to  average  volt- 
age when  the  curve  is  sinusoidal  is,  as  already  shown, 


v/2  7r 


Hence,  the  value  of  k is  1.11. 

The  foregoing  value  of  k is  computed  on  the  hypothesis  of  a 
uniform  magnetic  field  for  the  alternator,  corresponding  to  that 
shown  in  Fig.  12,  but  the  value  of  k in  commercial  alternators 
depends  upon  the  ratio  which  the  width  of  the  poles  bears  to 
the  polar  pitch  (distance  between  poles,  center  to  center),  the 
form  of  the  Polar  surfaces  and  the  distribution  of  the  magnet- 
ism thereover.  It  has  been  shown  * to  have  a minimum  limit  of 
unity  and  a maximum  limit  which  may  be  large,  depending 
upon  the  form  of  the  voltage  wave  set  up  in  the  individual 
armature  conductors. 


* Art.  11. 


THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


47 


20.  The  Effect  of  Distributed  Coils  on  Armature  Voltage.  — 

It  is  now  well  to  examine  the  relative  sizes  of  the  two  arma- 
tures compared  above,  and  the  effect  of  the  arrangements  of 
the  windings  upon  their  voltages. 

Thus  far  the  alternator  windings  have 
been  assumed  to  be  in  narrow  coils,  or 
arranged  so  that  at  each  instant  all  of 
the  conductors  are  equally  effective. 

This  is  not  the  case  in  actual  ma- 
chines, as  the  coils  must  have  apprecia- 
ble width.  If  two  collector  rings,  C and 
Cv  are  placed  upon  the  shaft  of  a direct- 
current  armature,  such  as  a Gramme 
ring  mounted  to  rotate  in  a bipolar 
field,  and  are  connected  to  armature 
conductors  which  are  in  opposite  coils 
(Fig.  30),  an  alternating  current  may 
be  taken  from  the  rings.  Such  an 
arrangement  is  called  a Double-current 
machine.  The  voltage  between  the 
rings  has  its  maximum  value  when  the 
conductors  connected  to  the  rings  are 
under  the  points  of  commutation,  and 
the  voltage  is  zero  when  the  armature 
has  revolved  90°.* 

The  maximum  of  the  alternating 
voltage  is  manifestly  equal  to  the 
direct  voltage  for  which  the  armature  was  designed,  and  the 
effective  alternating  voltage  is  .707  times  the  direct  voltage 
if  the  field  is  uniform  and  the  voltage  wave  sinusoidal.  In 
commercial  machines  the  field  is  not  uniform,  but  it  is  not 
likely  to  be  sufficiently  irregular  to  materially  disturb  the  ratio 
of  the  direct  and  effective  alternating  voltages  when  such  an 
armature  is  carrying  little  current.  When  the  armature  carries 
considerable  current,  armature  reactions  may  disturb  the  rela- 
tions to  a greater  or  less  degree.  An  armature  thus  arranged, 
with  the  conductors  uniformly  distributed  over  its  surface,  there- 
fore gives  substantially  a sine  alternating  voltage,  and  the  value 
of  k would  be  1.11  except  for  the  distribution  of  the  conductors 
* Jackson’s  Electromagnetism  and  the  Construction  of  Dynamos , p.  90. 


Two-pole  Field,  connected  to 
Collector  Rings. 


48 


ALTERNATING  CURRENTS 


over  the  armature  surface  ; but  the  effective  alternating  voltage 
is  only  .707  times  that  of  the  direct  voltage  developed  by  the 
same  conductors.  The  question  at  once  arises  as  to  the  cause 
of  this  loss  of  40  per  cent  or  more.  A little  consideration  shows 
that  the  Gramme  ring  acts  like  two  broad  coils  in  parallel,  which 
cover  the  whole  armature  and  unite  at  the  points  where  the  col- 
lecting rings  are  connected.  When  in  the  position  of  zero  vol- 
tage, the  two  halves  of  each  coil  are  so  located  in  the  field  as  to 
cut  lines  of  force  in  opposite  directions,  and  hence  the  voltage 
is  zero.  Similar  opposition  but  to  less  degree  is  found  in  the 
case  of  windings  which  cover  only  a portion  of  the  armature 
core,  the  extent  of  the  effect  depending  upon  the  ratio  of  the 
width  and  pitch  of  the  poles  to  the  width  of  the  coils.  It  is 
largely  avoided  if  the  coils  are  not  distributed  over  a greater 
width  than  the  distance  between  the  pole  tips.  Economy  of 
material  and  the  form  of  the  voltage  wave  are  also  factors  in 
determining  the  width  the  coils  should  have. 

On  account  of  the  differential  action  of  distributed  coils 
which  is  here  described,  it  is  necessary  to  include  another  con- 
stant in  the  formula  for  the  voltage  developed  by  alternating- 
current  armatures.  Calling  this  constant  k \ and  replacing  the 
product  k'k , by  AT,  the  formula  for  the  voltage  of  an  alternator 
armature  becomes 

KpS&V  _2KS<S>f 
~ p'  x 10s  x 60  ~ p'  x 10s’ 

Kapp  states  that  K varies  from  .29  to  1.15,*  but  in  the  greater 
number  of  commercial  cases  it  is  between  1.00  and  1.11. 

Instead  of  having  the  windings  laid  together  upon  the  surface 
of  the  armature,  it  is  usual  to  have  them  laid  in  slots  as  shown 
in  Fig.  31,  where  the  winding  is  distributed  in  four  slots  per 
pole.  If  the  winding  is  progressive,  as  in  the  Gramme  ring,  the 
wires  in  different  slots  will  set  up  voltage  waves  with  their  max- 
ima separated  by  the  angular  distance  between  the  centers  of 
the  slots,  as  illustrated  in  Fig.  32,  where  each  slot  is  supposed 
to  contain  the  same  number  of  conductors,  and  the  whole  width 
covered  by  the  winding  is  equal  to  one  half  the  pitch.  The 
resultant  curve,  due  to  connecting  the  conductors  all  in  series, 
may  be  found  by  adding  the  instantaneous  ordinates  together 

* Kapp’s  Dynamos,  Alternators,  and  Transformers,  p.  374. 


THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


49 


at  each  abscissa.  If  the  individual  curves  are  sinusoids,  the 
summation  is  of  this  form  : 


where  e is  the  instantaneous  voltage  of  the  resultant  curve, 
esm  is  the  maximum  ordinate  of  any  one  of  the  individual 
curves,  a is  the  angular  distance  from  the  point  where  the  first 


Fig.  31.  — Multipolar  Armature  with  Winding  distributed  in  Four  Slots.  Dotted 
Lines  show  Cross  Connections  at  the  Rear. 

curve  cuts  the  x axis,  /3  is  the  angular  distance  between  slots, 
center  to  center,  measured  in  electrical  degrees,  and  n is  the 
number  of  individual  curves  composing  the  resultant  curve. 
By  trigonometry  this  summation  becomes* 


For  the  purpose  of  finding  the  value  of  a.  at  which  the  resultant 
curve  has  its  maximum  height,  the  first  derivative  of  this 
function  with  respect  to  « may  be  equated  to  zero  and  reduced 
to 


e = esm\  sin«  + sin  (a  + /3)  + sin(«  + 2 /3)  + ••• 
+ sin  (a  + O — l)/3)b 


cos  a cos 


n — 1 


o 


or, 


* Hobson's  Plane  Trigonometry,  2d  ed.,  p.  89. 


50 


ALTERNATING  CURRENTS 


From  this  the  value  of  « required  to  give  the  maximum  ordi 
nate  of  the  resultant  curve  is  found  to  be 


« 


m 


7 T 

2 


/3, 


Fig.  32.  — Curve  of  Voltage  set  up  in  the  Separate  Slots  of  Progressive  Distributed 
Windings,  and  the  Resultant  Total  Voltage. 


and  substituting  this  value  of  « in  the  formula  for  e gives 


n — 1 a , 71  — 1 
x P + — — - 


a \ ■ 

p ) sin  — y cosec 


= e. 


sin  (71  3/-) 
sin  (fi/2)  ' 


i 


7 T 


THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


51 


The  effective  voltage  is  therefore 

sin  Q/3/ 2). 
- yo  sin  (/3/2)  ' 


If  E'  = — is  the  voltage  that  would  have  been  set  up  if  all 
the  wires  had  been  in  a single  narrow  slot,  the  formula  becomes 


E'  sin  (n/3/2) 
n sin  (/3/2)  ’ 

but  E'  is  the  voltage  given  by  the  formula  in  Art.  18 ; hence, 


2.22  ;SW  sin  Q/3/2) 

p'  x 108  n sin  (/3/2) 


Evidently  K of  Art.  20,  in  this  case,  is  equal  to  , 

J > n w sin  (/3/2) 

k and  k'  being  respectively  equal  to  1.11  and  s^n . in 

61  J ^ n sm  (/S/2) 

any  instance  in  which  the  conductor  voltages  are  of  sinusoidal 

form. 

Suppose  there  are  four  slots  equally  distributed  through  90 
electrical  degrees  in  an  alternator  armature  with  the  conductors 
of  one  coil  equally  distributed  in  those  slots  ; then  n = 4 and 
/3  = 221°,  whence 


k'  = \ sin  45°  cosec  11°  15'  = .906  (approx.) 

and  AT=  1.11  x .906  = 1.006  (approx.),  provided  the  con- 
ductor voltages  are  of  sinusoidal  form.  Suppose  now  that 
there  are  a total  of  500  active  conductors  in  series,  and  equally 
distributed  in  the  slots,  2,000,000  lines  of  force  per  pole,  and 
that  the  frequency  is  60  periods  per  second,  then 


F_  2 KS$>f  _ 2 x 1.006  x 500  x 2,000,000  x 60  _ 12n7  j 
p'  x 108  1 x 108 

Figure  31  shows  the  slots  for  a winding  such  as  contemplated 
in  the  problem  above,  and  Fig.  32  shows  the  voltage  curves. 

The  following  table  gives  the  value  of  K for  various  propor- 
tions of  the  armature  core  covered  with  winding  and  various 
numbers  of  slots  per  pole  up  to  four,  assuming  sinusoidal  con- 
ductor voltages. 


52 


ALTERNATING  CURRENTS 


TABLE  I 

Values  of  K for  Progressively  Distributed  Windings 


Group  of  Slots 

LOCATED  IN 

Number  of  Slots  in  a Group 

i 

2 

3 

4 

45° 

l.n 

1.09 

1.085 

1.08 

60° 

l.n 

1.075 

1.07 

1.065 

90° 

l.n 

1.025 

1.015 

1.01 

135° 

l.n 

.925 

.895 

.885 

180° 

l.n 

.78 

.735 

.725 

In  a single  phase  winding,  i.e.  an  armature  having  a single 
winding,  any  angular  width  for  the  group  of  slots  may  be  used, 
though  about  90°  is  common.  Where  there  are  two  independ- 
ent circuits  on  the  armature,  90°  is  also  used ; while  in  three- 
phase  circuits,  where  there  are  three  independent  circuits,  60° 
is  common.* 

The  values  given  for  K in  the  table  can  be  used  with  suffi- 
cient accuracy  for  most  commercial  forms  of  alternators,  as  the 
deviations  of  the  curves  from  sinusoidal  form  are  usually  not 
great  enough  to  change  the  constant  very  much ; but  this  con- 
dition cannot  be  relied  on  unchallenged.! 

When  desired,  the  curves  of  conductor  voltage  may  be  plotted, 
and  the  ordinates  of  the  voltage  curve  of  the  machine  may  be 
obtained  by  adding  corresponding  ordinates  of  the  conductor 
voltage  curves. 

The  vector  relations  of  sinusoidal  conductor  voltages  and  the 
machine  voltage  are  illustrated  in  Fig.  33,  where  OA , AA' , A' A", 
and  A"S  are  the  voltages  developed  by  the  conductors  in  the 
several  slots  of  the  foregoing  example.  These  are  laid  off  in  a 
vector  polygon.  The  effective  voltage  produced  in  the  con- 
ductor (or  conductors)  of  each  slot  is  333  volts,  and  the  resultant 
of  the  four  voltages  is  1207  volts  as  already  computed  by  the 
formula.  These  vectors  are  plotted  in  their  proper  relative 
positions.  Their  lengths  are  laid  down  equal  to  their  respect- 

* Art.  28. 

t De  la  Tour’s  Moteurs  Asynchronous  Polyphases,  Chap.  2. 


THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


53 


ive  effective  values.  The  corresponding  instantaneous  values  for 
the  instant  corresponding  to  Fig.  33  are  proportional  to  the  pro- 
jections of  the  various  vectors  on  the  vertical  axis.  The  reason 
that  the  coefficient  k'  is  less  than  unity  is  plainly  shown  by 
this  figure,  in  which  it  is  seen  that  the  component  or  conductor 


Fig.  33.  — Graphical  Representation  of  Voltage  generated  in  an  Armature  having 

Four  Slots. 


voltages  are  in  different  phases  and  their  resultant  is  therefore 
less  in  magnitude  than  their  arithmetical  sum. 

The  complex  expression  for  these  vectors  is 

OS  = COS")  +j(OS')  = [(06)  + (bb1)  + (b'b")  + (b"S")] 

+ j [(  Oa)  + (aa')  + (a' a")  + (a"  S')], 
or,  ( OS")  = (Ob)  + (bb')  + (b'b")  + (b"S"), 

and  ( OS')  — ( Oa)  + (aa')  -f  (a' a")  + (a" S'). 

Having  thus  obtained  the  numerical  values  of  the  terms  of 
the  complex  expression  for  the  resultant,  the  numerical  or  scalar 
value  of  the  resultant  is  obtained  from  the  relations 


( os)  = V(  os'y  + ( osy. 


54 


ALTERNATING  CURRENTS 


The  value  of  Oa,  aa\  Ob , bb',  etc.,  may  readily  be  found  by  use 
of  the  trigonometrical  table  ; for  instance 

AA'  = ( bb ')  +j  (aar)  = AA'  [cos  («  + 221°)  + j sin  (a  + 221°), 

and  therefore,  as  a,  in  this  case,  at  the  instant  indicated  in 
Fig.  33,  is  Ilf, 

(bb')  = 333  cos  33|°  and  (W)  = 333  sin  33|°. 

A winding  which  creates  several  equal  elementary  voltage 
curves  following  each  other  at  equal  angles,  but  added  to  obtain 
a resultant  at  the  collector  rings,  is  called  a Distributed  winding. 
Such  windings  are  shown  in  Figs.  67-70,  Chapter  III. 

Prob.  1.  In  an  alternator  armature  there  are  five  slots  in 
a group  and  the  angle  covered  by  them  is  120°.  What  is  the 
value  of  the  constant  K for  the  group  ? 

Prob.  2.  The  effective  voltage  set  up  in  the  conductors  in 
each  slot  of  problem  1 is  50  volts.  Determine  graphically  the 
total  voltage  of  the  machine. 

Prob.  3.  Obtain  the  answer  to  problem  2 by  the  method  of 
complex  quantities. 

Prob.  4.  An  armature  has  four  slots  in  a group  covering 
60°.  The  conductors  in  each  slot  develop  400  volts.  Determine 
the  total  voltage  developed  by  each  group,  when  the  conductors 
are  all  connected  in  series,  both  graphically  and  by  the  method 
of  complex  quantities. 

21.  Most  Economical  Width  of  Pole  Face.  — The  output  of  an 
alternator  is  proportional  to  the  product  £<!>  = swbw where  s 
is  the  number  of  conductors  per  unit  width  of  coil,  b the  number 
of  lines  of  force  per  unit  width  of  pole  face,  and  w and  w'  are 
respectively  the  widths  of  coil  and  pole  face.  In  order  to 
economize  material,  the  distance  between  pole  tips  may  be  taken 
as  equal  to  the  width  of  the  coil,  for  the  reason  explained  in  the 
previous  article,  and  this  makes  w + tv'  equal  to  the  pitch  of 
the  poles,  which  is  constant ; this  makes  the  product  ww\  and 
hence  the  output  of  the  machine,  a maximum  when  w is  equal 
to  w\  or  when  the  width  of  coil  and  pole  face  are  each  equal  to 
half  the  pitch.  The  result  thus  derived  must  be  modified  to 
suit  practical  conditions,  since  fringing  tends  to  increase  the 
width  of  field,  and  armature  reactions  tend  to  crowd  the  field 


THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


55 


towards  the  trailing  pole  tip,  thus  narrowing  the  field.  Experi- 
ment has  shown  that  it  is  best  to  have  the  coils  somewhat  wider 
than  half  the  pitch,  and  it  is  often  advantageous  to  have  the 
poles  slightly  less  than  half  the  pitch  in  width. 

22.  Polyphase  Alternators.  — As  has  been  shown,  only 
slightly  over  one  half  the  armature  surface  is  utilized  in  the 


machines  of  the  type  which  has  been  dealt  with,  and  which  are 
called  Single-phase  machines  or  Single-phasers  from  the  fact 
that  they  give  out  a single  current. 


A second  set  of  windings  may  be  placed  between  the  first,  as 
in  Fig.  34,  which  will  give  out  from  a second  pair  of  collector 
rings  or  terminals  another  alternating  current  displaced  90°  from 
the  first.  This  is  indicated  in  Fig.  35,  where  A,  A\  and  B , B', 
represent  the  two  currents.  Such  a machine  is  called  a Two- 
phaser.  Figures  36  and  37  show  by  diagram  two  methods,  and  c,  c' 


56 


ALTERNATING  CURRENTS 


of  connecting  up  the  coils  of  two-pkasers.  In  the  former  the 
phases  are  joined  together  at  a central  point  called  a Neutral 


ARMATURE 

TERMINALS 


Fig.  3(3.  — Two-phase  Armature. 
Four  Terminals. 


ARMATURE 

TERMINALS 


Fig.  37.  — Two-phase  Armature. 
Four  Terminals. 


ARMATURE 

TERMINALS 


Fig. 


38.  — Two-phase  Armature. 
Three  Terminals. 


point ; while  in  the  latter  the  two  phases  are  entirely  inde- 
pendent. The  figures  indicate  a simple  bipolar  ring,  to  avoid 

the  complexity  that  arises  in 
diagraming  the  windings  for 
multipolar  machines,  and  rep- 
resent either  a stationary  or 
rotating  armature.  Figure  38 
shows  the  coil  terminals  passing 
to  only  three  armature  termi- 
nals, which  is  made  possible  by 
utilizing  one  wire  as  a common 
conductor  for  both  circuits. 
The  current  in  the  common 
wire,  as  will  be  shown  later,*  is  V2  times  the  current  in  either 
of  the  independent  wires  when  the  currents  are  equal  in  the 
two  circuits  and  are  of  90°  difference  of  phase,  which  is  the 
condition  of  a balanced  two-phase  system;  and  the  voltage  be- 
tween the  two  independent  wires  is  then  V2  times  the  voltage 
between  either  of  those  and  the  common  wire. 

Instead  of  two  independent  circuits  being  placed  on  the  arma- 
ture, the  winding  space  may  carry  three  circuits,  as  illustrated 
in  Fig.  39,  and  the  machine  is  then  a Three-phaser.  The  three- 
pkaser  gives  three  voltage  curves  ordinarily  differing  from  each 
other  in  phase  by  120°  as  indicated  in  Fig.  40,  where  A.  A',  B . 
B\  and  C,  O’ , are  the  three  curves.  Figures  41  and  42  illustrate 
such  a winding  for  a simple  Gramme  ring  in  which  a , a',  b , b\  c,  c\ 

* Art.  100. 


THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


57 


represent  the  coils  of  each  phase  or  winding.  In  both  figures 
the  windings  are  so  connected  together  that  only  three  arma- 


ture terminals  are  used  instead 
of  three  independent  pairs.  A 
simple  diagram  of  the  connec- 
tion of  the  coils  of  Fig.  41  is 
shown  in  Fig.  43.  This  wind- 
ing is  called  a Mesh  or  Delta(A) 
winding.  The  currents  which 
combine  from  two  coils  at  the 
armature  terminals  (A,  B , (7) 
are  V3  times  greater  than  the 
current  in  a single  phase,* 


Fig.  41.  — Three-phase  Winding.  Mesh 
Connection. 


* Art.  100. 


58 


ALTERNATING  CURRENTS 


provided  the  currents  are  equal  in  the  three  windings  and  are 
120°  apart  in  their  phases,  which  is  the  condition  of  a balanced 
three-phase  system. 

The  arrangement  shown  in  Fig.  42  is  called  a Star  or  Wye  (Y) 
winding,  and  the  connection  of  the  coils  is  diagrammed  in  Fig. 

44.  The  voltage  measured 
between  any  two  armature 
terminals  connected  to  the 
Y winding  is  V3  times  the 
voltage  in  the  winding  of  a 
single  phase,*  provided  the 
voltages  are  equal  in  the  three 
windings  and  are  120°  apart 
in  their  phases. 

Machines  having  more  than 


Fig.  42.— Three-phase  Winding.  one  phase  or  winding  are 

St.u  Connection.  called  in  general  Polyphase 

and  sometimes  Multiphase  machines,  or  Polyphasers  or  Multi- 
phasers. 

The  methods  of  connecting  up  two-phase  armature  windings 
are  very  simple.  If  the  armature  is  wound  with  independent 
concentrated  coils,  each  coil  may  be  con- 
nected to  its  individual  armature  termi- 
nals, in  which  case  four  terminals  are  re- 
quired and  the  two  circuits  are  entirely 
independent ; or,  one  end  of  each  coil  may 
be  connected  to  a common  armature  ter- 
minal and  the  other  ends  to  independent 
terminals,  in  which  case 
but  three  terminals  are 
required  and  the  circuits  have  a common 
point.  If  the  armature  has  a direct-current 
or  closed-circuit  distributed  winding,  such 
as  is  treated  in  Art.  20,  it  may  be  made 
into  a two-phaser  by  connecting  collector 
Fig.  44.  - Diagram  of  Y pings  to  the  windings  at  the  ends  of  two 

Connection.  .. 

diameters  which  are  90  apart,  it  the  ma- 
chine is  bipolar  ; if  the  machine  is  multipolar,  the  connections 
must  be  made  as  described  in  Art.  28.  Four  rings  are  nec- 


Fig.  43. — Diagram  of  A 
Connection. 


* Art.  100. 


THE  VOLTAGE  DEVELOPED  BY  ALTERNATORS 


59 


essary  in  this  case,  as  the  use  of  a common  ling  would  cause 
the  permanent  short-circuiting  of  one  quarter  of  the  armature. 
A direct-current  armature  converted  into  a polyphaser  (of  any 
number  of  phases)  in  this  manner  has  a capacity  when  used 
as  a polyphase  alternator  which  is  nearly  equal  to  its  direct- 
current  capacity,  though  it  has  already  been  shown  that  its 
capacity  as  a single-phaser  is  only  seven  tenths  as  great  as  its 
direct-current  capacity.* 

The  manner  of  connecting  three-phase  armatures  is  not  so 
immediately  evident,  but  is  perfectly  simple.  It  is  illustrated 
in  Figs.  64  and  65.  Three  armature  terminals  are  universally 
used,  and  if  the  armature  is  wound  with  three  independent 
coils,  these  may  be  connected  to  the  terminals  in  either  of  two 
ways : (1)  one  end  of  each  of  the  coils  may  go  to  a common 
point,  and  the  other  ends  go  to  independent  terminals  ; or  (2) 
the  coil  ends  may  be  connected  together  two  and  two,  form- 
ing a sort  of  triangle,  and  connections  be  carried  to  the  arma- 
ture terminals  from  these  points.  The  latter  arrangement  makes 
each  coil  terminate  at  each  end  in  an  armature  terminal,  and, 
since  there  are  six  coil  ends  and  three  terminals,  each  terminal 
is  connected  with  two  coil  ends.  These  two  arrangements  are 
illustrated  respectively  in  Figs.  65  and  64,  each  of  which  shows 
the  winding  diagrammatically  as  developed  and  as  projected. 
The  following  considerations  make  it  perfectly  easy  to  connect 
the  coils  in  the  proper  order:  Fig.  40  shows  that  when  the  cur- 
rent in  one  coil  is  at  its  maximum  point,  the  currents  in  the 
other  two  are  equal  to  each  other  and  opposite  to  the  direction 
of  the  first ; then,  considering  the  instant  at  which  the  conduct- 
ors of  one  coil  (such  as  C in  the  figure)  are  directly  under  the 
poles,  if  we  connect  its  positive  end  to  the  common  or  neutral 
point  the  negative  ends  of  the  other  two  coils  must  be  con- 
nected to  the  same  point.  Each  of  the  free  ends  may  then  be 
connected  to  one  of  the  armature  terminals  and  the  connection 
is  completed  according  to  the  first  or  Y arrangement  (Fig.  65). 
To  make  the  second  or  A arrangement,  the  coils  must  be  con- 
nected so  that,  at  the  instant  considered,  the'  current  flows  by 
two  paths  through  the  armature  from  the  negative  to  the  posi- 
tive end  of  the  first  coil  (coil  C of  the  figure).  Consequently 
the  negative  ends  of  the  first  and  second  coils  go  to  one  arma- 

* Art.  20. 


60 


ALTERNATING  CURRENTS 


ture  terminal,  the  positive  ends  of  the  first  and  third  coils  to 
another  terminal,  and  the  free  ends  of  the  second  and  third  coils 
to  the  third  terminal  (Fig.  64). 

If  the  armature  has  a closed-circuit  or  distributed  winding, 
the  connection  is  very  simple,  as  the  rings  (terminals)  are  con- 
nected to  points  in  the  windings  120  electrical  degrees  apart. 
If  the  machine  is  multipolar,  it  must  be  remembered  that  one 
cycle,  or  360  electrical  degrees,  is  comprised  within  the  space 
of  twice  the  polar  pitch. 

In  a three-phase  machine,  if  the  armature  is  A connected, 
the  voltage  between  any  two  armature  terminals  is  equal  to  the 
voltage  developed  in  one  coil,  while  the  current  leaving  a ter- 
minal is  composed  of  the  vector  currents  in  two  coils;  and,  if 
the  armature  is  Y connected,  the  voltage  between  the  terminals 
is  composed  of  the  vector  voltages  developed  in  two  coils,  while 
the  current  leaving  a terminal  is  equal  to  the  current  in  a coil. 
The  line  currents  with  a balanced  load  are  V3  times  the  current 
in  a single  coil  in  the  A connection,  and  the  voltages  between 
lines  or  terminals  are  V3  times  the  voltage  of  a single  coil  in 
the  Y connection,  so  that  the  capacity  of  a machine  is  independ- 
ent of  the  way  in  which  its  armature  is  connected,  but,  for  a 
given  voltage  and  output,  the  windings  will  differ  (though  the 
weight  of  copper  will  be  the  same)  for  the  two  arrangements.* 

Prob.  1.  100  volts  are  set  up  in  each  winding  of  a balanced 
Y connected  three-phase  alternator  armature.  What  are  the 
voltages  between  the  terminals  of  the  machine  ? 

Prob.  2.  200  volts  are  generated  in  each  coil  of  a balanced 
A connected  three-phase  alternator  armature.  What  are  the 
voltages  between  the  terminals  of  the  machine  ? And  when 
50  amperes  flows  in  each  coil,  what  is  each  line  current? 

Prob.  3.  The  two  windings  of  a balanced  two-phase  armature 
are  connected  to  three  line  wires.  The  voltage  developed  in 
each  winding  is  100  volts;  the  line  wires  are  designated  A,  B.  C , 
where  B is  the  common  return  wire.  What  are  the  voltages 
between  A and  74,  B and  (7,  and  A and  C ? 

Prob.  4.  In  problem  4 the  current  flowing  in  each  phase  is 
50  amperes.  What  is  the  current  in  the  common  return  wire  ? 

* Chap.  VIII. 


CHAPTER  II 


ELEMENTARY  STATEMENTS  CONCERNING  TRANSFORMERS 
AND  MEASURING  INSTRUMENTS 


23.  The  Fundamental  Principle  of  the  Transformer.  — If  a coil 
of  wire  is  wound  upon  a closed  iron  core  of  high  magnetic  con- 
ductivity, i.e.  low  reluctance,  as  illustrated  in  Fig.  45,  a small 
current  passed  through 
the  coil  will  produce  a 
powerful  magnetic  flux. 

If  an  alternating  current 
is  passed  through  the  coil, 
the  magnetic  flux  changes 
with  the  cycles  of  the  cur- 
rent and  the  chaneino-  ^ig.  45. — Coil  Wound  upon  an  Iron  Core  of  Low 
’ , ..  ‘ ° ° Reluctance, 

number  ot  lines  ot  force 

linked  with  the  coil  create  a Counter  voltage  in  the  coil  which 
opposes  the  tendency  of  the  voltage  impressed  at  the  terminals 
to  send  current  through  the  coil.  This  action  is  in  accord  with 
a natural  law  which  may  he  stated  as  follows  : When  a current 

in  a conductor  is  caused  to  change  in  value , or  a conductor  is 
moved  in  a magnetic  field  so  as  to  cut  lines  of  force , an  induced 

electro-motive  force  is  set  up  in  the 
conductor  which  tends  to  oppose  the 
change  in  current  value  or  the  move- 
ment of  the  conductor.  This  state- 
ment is  a corollary  of  the  Laiv  of 
the  Conservation  of  Energy. 

If  another  coil  is  wound  upon 
the  same  core,  as  illustrated  in 
Fig.  46,  the  magnetic  flux  set  up 
by  current  in  the  first  coil  passes 
through  the  second  coil  also,  and 
the  changes  in  the  strength  of  the 
field  set  up  an  induced  voltage  in  this  second  coil  similar  to 
that  in  the  first.  An  Induction  coil  of  this  character,  used  with 

61 


Fig.  46.  — Low  Reluctance  Iron 
Core  carrying  Primary  and  Sec- 
ondary Coils. 


62 


ALTERNATING  CURRENTS 


alternating  currents,  is  called  a Transformer.  The  coil  which 
receives  the  current  from  an  outside  source  is  called  the 
Primary  coil  and  that  in  which  voltages  are  induced  by  the 
magnetic  flux  set  up  by  current  in  the  primary  coil  is  called 

the  Secondary  coil. 

The  phenomenon  is  called  Self-induction  when  electro-motive 
force  is  induced  in  a coil  as  the  result  of  changes  of  the  current 
in  its  own  conductors ; and  the  phenomenon  is  called  Mutual- 
induction  when  an  electro-motive  force  is  induced  in  a coil  as 
the  result  of  changes  of  the  current  flowing  in  a neighboring 
coil.  When  an  alternating  current  flows  through  a primary 
coil,  it  is  obvious  that  the  magnetic  flux  set  up  in  the  iron  core, 
such  as  is  illustrated  in  the  transformer  of  Fig.  46,  must  also 
be  alternating  and  of  the  same  period ; and  that  the  induced 
voltages  in  both  the  coils  are  also  alternating  and  of  the  fre- 
quency of  the  primary  current. 

The  maximum  voltage  induced  per  turn  of  wire  in  either  coil 
is  proportional  to  the  maximum  value  of  the  alternating  mag- 
netic flux  which  links  the  coil  and  to  the  rate  of  change  of  the 
flux.  The  latter  is  proportional  to  the  frequency  of  the  current. 
Therefore,  if  all  the  turns  of  each  coil  are  in  series  and  the 
same  magnetic  flux  links  the  two  coils,  the  voltages  induced  in 
the  primary  and  secondary  coils  are  respectively  proportional 
to  their  number  of  turns  ; or 


where  R[  and  U'2  are  the  induced  voltages  and  nv  n2  are  the 
turns  in  the  primary  and  secondary  coils,  respectively. 

In  the  case  of  commercial  transformers,  when  no  current  is 
flowing  in  the  secondai-y  coil,  the  self-induced  voltage  in  the 
primary  coil  almost  equals  the  voltage  impressed  at  its  termi- 
nals. This  is  because  only  a very  small  Exciting  current  is 
required  for  the  core  magnetization  necessary  to  set  up  an 
induced  voltage  equal  to  the  impressed  voltage,  and  the  voltage 
used  in  sending  the  current  through  the  resistance  of  the  con- 
ductors of  the  primary  coil  is  practically  negligible.  The  in- 
stantaneous value  of  counter  voltage  at  each  instant  must  be 
smaller  than  the  corresponding  instantaneous  value  of  the 
impressed  voltage  by  an  amount  equal  to  the  IR  drop  at  that 


ELEMENTARY  STATEMENTS 


63 


instant.  The  IR  drop  in  a commercial  transformer  is  usually 
very  small  when  the  secondary  circuit  is  open. 

It  may  therefore  be  said  that,  in  an  ordinary  unloaded  com- 
mercial transformer,  the  ratio  of  the  primary  impressed  voltage 
to  the  secondary  induced  voltage  is  substantially  equal  to  the 
ratio  of  the  numbers  of  turns  in  series  respectively  in  the 
primary  and  secondary  coils.  That  is,  substantially, 


where  R1  is  the  primary  impressed  voltage. 

The  induced  voltage  set  up  in  the  winding  of  the  secondary 
coil  is  obviously  in  phase  with  the  counter  voltage  of  the  pri- 
mary coil,  assuming  them  to  be  set  up  by  the  same  magnetic  flux, 
and  this  secondary  voltage  is  practically  opposite  to  the  voltage 
impressed  on  the  primary  coil.  Closing  the  circuit  of  the 
secondary  coil  through  resistance  (such  as  a number  of  incandes- 
cent lamps)  allows  a cui’rent  to  flow  under  the  impulse  of  the 
secondary  induced  voltage,  which  current  is  opposed  in  phase 
to  the  current  caused  by  the  impressed  voltage  to  flow  through 
the  primary  coil.  The  result  is  that  the  magnetizing  effect  of 
the  primary  current  is  opposed  by  the  magnetizing  effect  of  the 
secondary  current,  and  the  primary  current  must  increase  suf- 
ficiently to  overcome  the  magnetic  effect  of  the  secondary  cur- 
rent and  continue  to  magnetize  the  core  as  before.  The  ampere 
turns  of  the  primary  coil  must  therefore  increase  by  an  amount 
equal  and  opposite  to  the  ampere  turns  of  the  secondary  coil. 
That  is,  ... 

where  n2i2  are  the  instantaneous  ampere  turns  of  the  secondary 
coil,  n1i1  are  the  ampere  turns  in  the  primary  coil,  and  n^,u  rep- 
resents the  part  of  the  primary  ampere  turns  required  to  set  up 
the  magnetic  flux  in  the  core  at  the  corresponding  instant. 

The  current  in  the  primary  coil,  then,  must  be  slightly 

greater  than  2 i2  in  order  to  supply  the  core  magnetization, 

ni 

but  in  commercial  transformers  this  extra  magnetizing  current 
is  small,  and  may  usually  be  neglected  ; and  the  statement  may 
therefore  be  made  that  the  currents  in  the  primary  and  secondary 
coils  of  the  usual  commercial  transformers  designed  for  constant 


64 


ALTERNATING  CURRENTS 


voltage  service  are  almost  inversely  proportional  to  the  number 
of  turns  in  the  two  coils.  That  is,  approximately, 


When  currents  flow  in  the  secondary  coil,  the  voltage  lost  in 
the  resistance  of  the  coils  can  no  longer  be  neglected.  This 
drop  at  full  load  amounts  to  from  one  to  five  per  cent  of  the 
voltage  at  no  load  in  the  transformers  of  ordinary  practice,  and 
the  ratio  of  the  voltages  at  the  primary  and  secondary  terminals 
is  no  longer  correctly  represented  by  the  ratio  of  the  numbers 
of  primary  and  secondary  turns.  Magnetic  leakage,  whereby 
the  magnetic  flux  linking  one  coil  becomes  different  from  the 
flux  linking  the  other  coil,  also  affects  the  ratio  of  voltages. 
These  deviations  from  ideal  action  are  fully  discussed  in  a 
later  chapter. 

A transformer  may  be  defined  as  a combination  of  coils  and 
magnetic  core  for  transforming  alternating  currents  of  a given 

voltage  into  propor- 
tional alternating 
currents  of  another 
fixed  voltage  with- 
out the  intervention 
of  physical  motion. 
The  core  of  a trans- 
former must  be 
Laminated,  that  is, 
made  up  of  very 
thin  sheets  of  soft 
iron  or  steel  laid 
parallel  to  the  lines 
Figure  47  shows 


TERMINAL  OF 
PRIMARY 
W'NDING 


TERMINALS  OF 
SECONDARY 
WINDINGS 


TERMINAL  OF 
PRIMARY 
WINDING 


TERMINALS  OF 
SECONDARY 
WINDINGS 


Fig.  47.  — Skeleton  View  of  a Transformer. 


of  force  to  prevent  the  flow  of  eddy  currents 
a skeleton  view  of  a commercial  transformer. 

If  there  were  no  losses  of  power  in  the  operation  of  a trans- 
former, the  power  given  out  by  the  secondary  coil  would 
obviously  be  equal  to  the  power  received  by  the  primary  coil ; 
that  is,  P2  would  be  equal  to  Pr  But  an  actual  transformer 
cannot  be  operated  without  I2R  losses  in  the  conductors, 
hysteresis  losses  in  the  core,  and  eddy  current  losses  in  the 
core  and  perhaps  other  parts,  so  P2  is  always  smaller  than  Py 


ELEMENTARY  STATEMENTS 


65 


The  difference  is  only  a few  per  cent  of  the  input  in  the  case 
of  a well-designed  modern  constant  voltage  transformer  oper- 
ated at  normal  full  load. 

A modified  transformer  having  the  secondary  mounted  upon 
a rotating  core  and  so  arranged  as  to  deliver  mechanical  instead 
of  electrical  power  is  called  an  Induction  motor.  A discussion  of 
transformers  and  induction  motors  is  to  be  found  in  Chapter  XI. 

Prob.  1.  A certain  transformer  delivers  to  the  external 
secondary  circuit  a power  of  50  kilowatts.  If  the  efficiency 
of  the  transformer  is  96  per  cent  at  this  load,  what  power  must 
be  supplied  to  it  ? 

Prob.  2.  In  a fully  loaded  transformer  the  voltage  supplied 
is  1000  volts,  the  current  50  amperes,  and  the  secondary  voltage 
is  100  volts.  What  is  the  approximate  value  of  the  secondary 
current  ? 

24.  Alternating-current  Amperemeters,  Voltmeters,  and  Watt- 
meters.*— The  three  general  classes  of  direct-reading  instru- 
ments most  suitable  for  making  measurements  in  alternating- 
current  circuits  have  already  been  given. f Working  gear  for 
each  of  the  three  most  important  of  the  instruments  for  com- 
mercial electrical  measurements  (namely,  amperemeters,  volt- 
meters, and  wattmeters)  may  be  made  on  the  principles  of 
either  of  the  three  types  named  in  Art.  17.  Paragraphs  a,  d , 
and  e.  following,  deal  with  the  first  class. 

a.  Electro-dynamometers . — Large  numbers  of  the  best  known 
alternating-current  instruments  depend  upon  the  electro-dyna- 
mometer principle,  i.e.  the  magnetic  action  of  the  current  in 
one  coil  upon  the  magnetic  field  of  a current  in  another  coil. 
Such  instruments  must  be  constructed  without  masses  of  solid 
conducting  material  about  them,  or  eddy  currents  may  be  set 
up  and  the  readings  of  the  instruments  become  affected  by  the 
frequency  of  the  current  to  be  measured.  This  is  due  to  the 
dynamic  effect  which  eddy  currents  (induced  short-circuit 
currents)  circulating  in  the  metallic  masses  produce  upon  the 

* Jackson’s  Elementary  Electricity  and  Magnetism,  Chaps.  XIII,  XYI,  and 
§231. 

t Art.  17. 

F 


66 


ALTERNATING  CURRENTS 


currents  in  the  moving  parts  of  the  instruments.*  Likewise, 
iron  cores  can  be  used  only  with  the  utmost  caution  in  the 
construction  of  the  working  gear,  even  though  laminated  (built 
up  of  thin  wires  or  sheets)  to  lessen  eddy  currents,  as  the 
hysteresis  of  the  iron  is  likely  to  impair  the  accuracy  of  the 
readings.)  If  the  instrument  is  to  be  used  as  a voltmeter, 
the  self-induction  and  electrostatic  capacity  of  the  wdn dings 
must  be  reduced  to  negligible  values,  or  the  readings  may  be 
affected  by  the  circuit  frequency.  The  elimination  of  self- 
inductive  or  capacity  effects  from  the  voltage  coils  of  watt- 
meters is  also  very  important,  as  is  fully  demonstrated  later. ) 

If  any  of  these  harmful  elements  are  present,  the  instrument 
must  be  calibrated  under  the  exact  conditions  of  frequency, 
current,  or  power,  with  which  it  is  to  be  used,  or  the  readings 
may  prove  to  be  quite  erroneous.  On  the  other  hand,  such 

instruments,  property  constructed, 
can  be  relied  on  to  read  with  equal 
accuracy  in  alternating-current  cir- 
cuits having  any  of  the  commercial 
frequencies ; and  they  may  ordinarily 
be  calibrated  in  a direct-current  cir- 
cuit (unless  iron  is  used  in  the  work- 
ing gear).  Figure  48  illustrates  a 
long-used  form  of  electro-dynamom- 
eter arranged  for  use  as  an  ampere- 
meter. This  is  often  called  the 
Siemens  Electro-dynamometer . This 
form  gave  excellent  service,  though 
it  is  somewhat  clumsy.  One  coil.  F, 
in  this  instrument  is  fastened  to  the 
frame  of  the  instrument,  and  the 


Fig.  48.  — Siemens  Electro-dyna- 
mometer and  Diagram  of  Con- 
nections to  Circuit. 


other  coil,  M,  which  stands  at  right  angles  to  the  first,  is 
suspended  by  a heavy  silk  fiber,  so  that  it  is  free  to  move. 
The  ends  of  the  wire  composing  the  movable  coil  dip  into 
little  cups,  (7(7,  containing  mercury,  which  are  connected  with 
the  main  circuit  so  that  the  current  can  enter  and  leave  the 
coil.  The  movable  coil  is  attached  to  a spring  6r,  the  other 
end  of  which  is  connected  to  a thumbscrew  T,  called  a Torsion 


* Arts.  90  and  111. 


t Art.  106. 


I Art.  90. 


ELEMENTARY  STATEMENTS 


67 


head , by  means  of  which  the  spring  may  be  twisted.  When  a 
current  flows  in  the  coils,  the  magnetic  force  tends  to  turn  the 
movable  coil  around  so  as  to  place  it  parallel  with  the  fixed 
coil.  This  force  is  balanced  by  twisting  the  spring  by  means 
of  the  thumbscrew.  The  amount  of  twist,  as  shown  by  a 
pointer  B attached  to  the  screw,  is  proportional  to  the  force 
exerted  by  the  coils  on  each  other.  This  force  is  proportional 
to  the  square  of  the  current  flowing  in  the  coils,  since  the  coils 
are  connected  in  series  and  the  magnetism  set  up  by  each  coil 
is  proportional  to  the  current,  and  they  act  upon  each  other 
mutually.  The  pointer  N indicates  whether  the  movable  coil 
is  at  its  zero  position.  The  “ binding  posts  ” for  connecting 
the  instrument  into  circuit  are  shown  at  AA. 

According  to  the  known  laws  of  electro-dynamics,  the  torque 
acting  at  each  instant  to  rotate  the  movable  coil  is  proportional 
to  the  product  of  the  simultaneous  instantaneous  values  of  the 
currents  flowing  in  the  coils,  and  is  equal  to  this  product  multi- 
plied by  a constant  of  the  instrument  which  is  fixed  by  the  rela- 
tive dimensions,  the  numbers  of  turns,  and  the  relative  positions 
of  the  two  coils.  The  average  torque  acting  on  the  movable 
coil  through  each  period  of  an  alternating  current  is  therefore 
proportional  to  the  average  (taken  through  the  period)  of  the 
products  of  the  corresponding  instantaneous  currents  in  the 
two  coils ; and  when  the  two  coils  are  in  series  relation,  as  in 
the  electro-dynamometer  type  of  amperemeter  or  voltmeter, 
the  current  at  each  instant  is  the  same  in  the  two  coils,  and  the 
average  torque  is  proportional  to  the  average  of  the  squares  of 
the  instantaneous  values  of  the  current.  That  is,  the  average 
torque  is  equal  to  iT(av.  *2),  where  K is  the  constant  of  the 
instrument  fixed  by  the  construction.  This  average  torque  is 
balanced  by  the  spring  attached  to  the  torsion  head  in  the 
Siemens  electro-dynamometer;  and  if  the  spring  is  uniform, 
the  movement  of  the  torsion  head  required  to  hold  the  mova- 
ble coil  in  its  initial  position  is  proportional  to  (av.  «2)  = I2. 

Instruments  of  the  electro-dynamometer  type  are  more  con- 
venient if  the  movable  coil  carries  a pointer  which  moves  over 
a scale.  This  arrangement  has  largely  displaced  the  Siemen’s 
type.  Such  an  instrument,  of  Weston  make,  is  illustrated  in 
Figs.  49  and  50.  This  instrument  is  like  that  of  Fig.  48  in 
principle,  but  it  has  lighter  coils  connected  in  series  with  a large 


68 


ALTERNATING  CURRENTS 


non-inductive  resistance  JK  to  reduce  the  time  constant  ot  the 
instrument  circuit,  and  make  it  practicable  for  use  as  a volt- 
meter.* The  resistance  of  the  commercial  alternating-current 

voltmeters  of  this 
type  is  from  about 
20  to  40  ohms  per 
volt  of  the  maximum 
reading  of  the  scale. 
The  construction  of 
the  voltmeter  shown 
in  Figs.  49  and  50, 
which  are  lettered 
alike,  is  as  follows : 
B is  the  movable 
coil,  and  AA  is  the 
stationary  or  fixed 
coil ; AT,  iVT,  0 are 

Fig.  49.  — Perspective  View  of  Interior  of  Weston  Alter-  ]jin(^pn(-r  posts  • J 
nating-current  Voltmeter  from  below.  e I ‘ ’ ’ 

K are  extra  resist- 
ance coils,  wound  non-inductively  \ L is  a special  variable 
resistance  used  to  correct  the  readings  for  variations  of  tem- 
perature ; D is  a push-button  switch  ; C, , C1  are  springs  through 
which  the  current  en- 
ters and  leaves  the 


movable  coil ; P is  the 
needle  or  pointer  ; H is 
the  scale,  which  is  en- 
graved with  two  sets  of 
figures;  and  6r  is  a ther- 
mometer with  its  bulb 
near  the  coils  of  the  in- 
strument, and  its  stem 
in  view  near  the  scale 
of  the  instrument.  The 
instrument 

boxed  up  so  that  only 
the  scale  H , IT,  over  which  the  pointer  moves,  the  end  of  the 
pointer,  the  dial  of  L , and  the  stem  of  the  thermometer  are 


is  usually  Fig.  50.  — Diagram  of  Weston  Alternating-current 

Voltmeter. 


* Jackson's  Electricity  and  Magnetism,  pp.  250,  251. 


ELEMENTARY  STATEMENTS 


G9 


visible.  The  thermometer  is  not  always  essential  when  the  best 
modern  resistance  material  of  low  temperature  coefficient  is 
used. 

When  the  instrument  is  in  service,  the  voltmeter  is  connected 
to  the  circuit  by  means  of  the  binding  post  0 , and  either  the 
binding  post  M or  the  binding  post  N.  When  the  button  D is 
depressed,  current  flows  through  the  fixed  and  movable  coils, 
and  through  the  resistance  J -f  K or  J alone  (depending  upon 
which  binding  post,  M or  W,  is  used),  and  the  pointer  is  made 
to  move  over  the  scale  by  the  movement  of  the’  movable  coil, 
which  action  is  caused  by  the  electro-magnetic  attractions  be- 
tween the  current  in  its  windings  and  the  current  in  the  wind- 
ings of  the  fixed  coil. 

A pointer  on  the  dial  L is  set  at  a mark  which  corresponds 
to  the  temperature  indicated  by  the  thermometer,  and  more  or 
less  of  the  resistance  coils  connected  to  the  dial  are  thus  in- 
cluded in  the  voltmeter  circuit.  In  this  way  the  resistance  of 
the  voltmeter,  measured  from  binding  post  to  binding  post,  may 
be  kept  uniform,  regardless  of  the  temperature  of  the  instru- 
ment, and  the  readings  are  thus  corrected  for  the  variations  of 
temperature.  The  resistance  of  the  windings  AA  and  B , added 
to  the  resistance  of  J",  is  just  equal  to  one  half  of  the  resistance  of 
the  same  windings  plus  the  resistance  of  J -\-  K.  Consequently 
only  half  the  voltage  between  the  instrument  terminals  is  re- 
quired to  cause  a given  movement  of  the  needle  when  the  bind- 
ing posts  0 and  N are  used,  as  when  the  binding  posts  0 and 
M are  used.  The  scale  which  reads  up  to  7.5  volts,  in  the 
instrument  illustrated,  is  therefore  used  in  connection  witli 
binding  posts  0 and  W,  and  the  scale  which  reads  up  to  15  volts, 
with  the  binding  posts  0 and  M.  The  movement  of  the  coil 
and  pointer  is  opposed  by  the  springs  C,  Cv  and  the  scale  is 
engraved  so  that  the  instrument  is  direct  reading.  As  the 
springs  offer  an  opposing  force  which  is  practically  proportional 
to  the  extent  of  movement  of  the  movable  coil,  and  the 
torque  acting  to  move  the  movable  coil  is  proportional  to  the 
square  of  the  current,  it  is  obvious  that  the  scale  of  the  instru- 
ment must  widen  out  from  left  to  right.  That  is,  the  scale  is 
a “ square-root  scale,”  but  by  proper  design  of  the  coils  of  such 
instruments  it  is  possible  to  make  the  widest  divisions  at  the 
most  important  part  of  the  scale. 


70 


ALTERNATING  CURRENTS 


Fig.  51.  — Diagram  of  Wattmeter  Connections. 


If  an  electro-dynamometer  is  to  be  used  as  a wattmeter,  the 
movable  coil  is  usually  placed  in  series  with  a large  non-induc- 
tive resistance  and  connected  across  the  circuit,  and  the  station- 
ary coil  is  of  low 
resistance  and  is  con- 
nected in  series  with 
the  circuit.  This  is 
illustrated  in  Fig.  51. 
Under  these  circum- 
stances the  instru- 
ment can  evidently 
be  calibrated  to 
measure  watts,  since 
the  torque  is  proportional  to  the  average  of  the  products  of 
the  instantaneous  voltages  with  the  corresponding  instanta- 
neous currents.  Figure  52  shows  a portable  form  of  wattmeter 
with  its  cover  removed  so  as 
to  show  the  working  parts, 
which  are  indicated  by  let- 
ters. MM  and  0 0 are  bind- 
ing posts,  the  former  of 
which  are  terminals  for  the 
“ voltage  coil,”  and  the  latter 
terminals  for  the  “ current 
coil  ” (one  of  the  latter  is  at 


the  far  corner  of  the  instru-  Fig.  52.  — Partial  Perspective  View  of  Hoyt 
ment  and  is  hidden);  A is 

the  stationary  or  fixed  coil;  BB  is  the  movable  coil:  P is  the 
pointer  or  “needle”;  H is  the  scale;  K is  a non-inductive 
extra  resistance  coil  which  is  placed  in  series  with  the  voltage 
coil;  B is  a torsion  head  by  means  of  which  the  spring  E may 
be  turned  so  as  to  bring  the  movable  coil,  which  is  attached  to 
it,  into  zero  position.  This  position  is  indicated  by  the  pointer 
P' . When  the  pointer  P'  points  to  zero,  the  pointer  _P,  which 
is  attached  to  the  torsion  head,  points  to  the  reading  of  the 
instrument.  R and  Q are  the  wooden  base  and  supports  of  the 
instrument.  CC  are  flexible  conductors  for  affording  the  cur- 
rent ingress  and  egress  to  and  from  the  movable  coil. 

Wattmeters  are  usually  made  so  that  the  needle  P is  directly 
attached  to  the  movable  coil  in  the  manner  illustrated  in  FJcr. 


ELEMENTARY  STATEMENTS 


71 


50;  and  in  that  case  the  flexible  conductors  CC  are  replaced 
by  conducting  springs,  also  as  illustrated  in  Fig.  50. 

Electro-dynamometers  are  made  with  an  endless  variety  of 
details.  One  type  of  special  accuracy  is  the  Kelvin  balance, 
which  is  frequently  used  as  a standard  for  calibrating  less  per- 
manent instruments.  Such  a balance,  illustrated  in  Fig.  53, 


consists  essentially  of  two  electro-dynamometers  which  tend  to 
turn  an  arm  pivoted  at  the  center  between  the  movable  coils. 
The  fixed  and  movable  coils  in  these  instruments  are  parallel 
to  each  other  and  horizontal.  The  two  movable  coils  are  con- 
nected in  circuit  in  series  with  each  other  so  that  the  current 
in  one  circulates  in  the  same  direction  as  the  current  in  the 
other.  This  balances  the  effect  of  any  reasonably  uniform 
external  magnetic  field  (such  as  the  earth’s  field)  on  the  bal- 
ance arm  and  eliminates  such  effects  from  the  readings,  which 
is  very  desirable  when  the  instrument  is  used  in  connection 
with  direct  currents.  When  the  instrument  is  used  with  alter- 
nating currents,  the  external  fields  would  not,  in  any  event, 
affect  the  readings  unless  these  fields  also  were  alternat- 
ing, and  of  the  same  frequency.  The  force  with  which  the 
movable  coils  tend  to  move  when  a current  flows  in  the  sets  of 
instrument  coils  is  directly  balanced  and  weighed  by  means  of 
a slider  A moving  on  a scale  beam  B.  Some  of  the  balances 
having  a large  number  of  turns  are  not  suitable  for  alternat- 
ing currents  on  account  of  the  coils  being  of  too  high  self- 
inductance. 

b.  Hot-wire  Instruments.  — If  a heated  wire  is  carefully 
inclosed  so  that  its  temperature  is  not  affected  by  air  currents, 
it  will  rise  a definite  number  of  degrees  in  temperature  for 
every  current  that  is  passed  through  it,  and  the  rise  is  propor- 


ALTERNATING  CURRENTS 


VI 


tional  to  the  energy  expended  in  the  wire  and  therefore  to  the 
square  of  the  current.  The  length  of  wire  increases  practically 
in  direct  proportion  to  its  rise  in  temperature  when  it  is  heated, 
and  the  length  again  decreases  when  the  wire  is  cooled.  Con- 
sequently, when  currents  of  different  strengths  flow  through  a 
wire,  it  will  take  up  a corresponding  length  with  each  current,  and 
measuring  its  length  therefore  measures  the  square  of  the  current. 

Hot-wire  instruments  evidently  average  up  the  instantaneous 
currents  squared,  since  the  heat  developed  in  the  wire  is  12B  and 
(the  wire  being  protected  so  that  the  coefficients  of  radiation 

and  convection  are  constant) 
the  temperature  is  proportional 
to  the  rate  at  which  heat  is  lib- 
erated in  the  wire.  Figrure  54 
shows  a hot-wire  switchboard 
instrument.  The  current  passes 
through  the  platinum  silver 
wire  A and  this  fine  wire  is  so  ar- 
ranged, by  the  use  of  special 
springs,  that  its  elongation  oper- 
ates the  pointer  B , which  is 
mounted  on  jewels.  The  wire  is 
of  too  low  resistance  to  adapt  the  instrument  for  direct  use  as 
a voltmeter  on  ordinary  voltages,  and  a resistance  coil,  non-in- 
ductively  wound,  is  put  inside  the  case  and  connected  in  series 
relation  with  the  wire  to  make  a voltmeter.  The  wire  is  of 
small  current-carrying  capacity  and  must  be  shunted  to  adapt 
the  instrument  for  use  as  an  amperemeter  unless  the  current  is 
only  a few  milliamperes.  Hot-wire  instruments  have  not  been 
used  commercially  as  wattmeters. 

A shunt,  except  for  very 
small  currents,  is  ordi-  a wire  ot 


Fig.  54.  — Hot-wire  Instrument. 


MAIN  CIRCUIT 


MILLI  AMMETER 


narily  placed  within  the 
case  and  permanently  at- 
tached in  most  types  of 
portable  amperemeters ; 
unless  the  instruments  are 
to  be  used  for  large  currents  or  on  switch-boards,  when  the 
shunt  is  separated  from  the  instrument  and  they  are  connected 
in  circuit  as  illustrated  in  Fig.  55,  or,  as  is  frequently  the  case,  a 


Fig.  55.  — Ammeter  with  Shunt. 


ELEMENTARY  STATEMENTS 


73 


current  transformer  takes  the  place  of  the  shunt.  The  shunt 
should  be  made  of  material  having  a low  temperature  coefficient 
and  large  surface  and  should  be  substantially  non-inductive. 

c.  Electrostatic  Instruments.  — Electrostatic  instruments  are 
built  on  the  principle  of  the  electrometer.  Figure  56  shows  the 
plan  of  a quadrant  electrometer.  If  the 
needle  (which  is  indicated  by  the  dotted  line 
in  the  illustration)  and  a pair  of  opposite 
quadrants  are  connected  to  a point  in  an  elec- 
tric circuit  and  the  other  pair  of  quadrants  to 
another  point  in  the  circuit,  the  needle  will 
tend  to  deflect  with  a force  proportional  to 

x A’  IKj.  tIU.  A 1 iX  11  UA 

the  square  of  the  difference  of  potential  be-  Quadrants  and 

tween  the  points.  This  must  be  so,  since  Needle  for  a Quad- 

, , ...  , • rant  Electrometer. 

the  force  tending  to  move  the  needle  is  pro- 
portional to  qq',  where  q and  q'  are  the  electric  charges  on  the 
two  elements  of  the  instrument  respectively,  and  since  both 
these  quantities  vary  with  the  difference  of  potential  between 
the  points  of  connection  with  the  circuit.  Evidently  qq'  may 

be  written  Aq 2,  where  A is  a constant 
depending  upon  the  shapes  and  dimen- 
sions of  the  parts  of  the  instrument.  If 
the  movement  of  the  needle  is  opposed 
by  a uniform  spring,  as  usual  in  elec- 
trical instruments,  the  deflections  will 
be  proportional  to  the  torque  and  there- 
fore the  readings  will  be: 

D = .5( av.  j2)  = _ST( av.  e2), 

where  B and  K are  constants.  Such 
an  instrument  can  therefore  be  cali- 
brated to  read  effective  volts,  hut  (like 
the  electro-dynamometer  voltmeter)  it 
has  a “square-root  scale.”  Figure  57 
shows  a Kelvin  electrostatic  voltmeter 
arranged  to  be  used  commercially.  It 
is  seen  from  the  figure  that  there  are 
a large  number  of  sets  of  quadrants 
and  needles,  one  above  the  other.  The  needles  receive  their 
charges  through  a fine  phosphor  bronze  wire  which  acts  as  a 


Fig.  57. — Kelvin  Electro- 
static Voltmeter. 


74 


ALTERNATING  CURRENTS 


supporting  filament  for  the  needles  and  also  furnishes  the  nec- 
essary restraining  force. 

If  an  instrument  of  this  class  is  to  he  used  as  an  ammeter,  it 
must  be  in  connection  with  a shunt,  though  it  is  not  well  suited 
for  this  purpose.  In  this  case  the  needle  deflects  proportionally 
to  the  square  of  the  voltage  drop  in  the  shunt,  and  hence  to  the 
square  of  the  current  flowing.  Special  arrangements  are  made 
when  electrostatic  instruments  are  applied  to  power  measure- 
ments, as  will  be  explained  later.* 

d.  Magnetic  Vane  Instruments. — Magnetic  vane  voltmeters 
and  ammeters  are  instruments  in  which  a very  light  vane  or 
needle  of  soft  iron  is  caused  to  turn  under  the  influence  of  a mag- 
netic field.  The  magnetic  vane  will  theoretically  have  a torque 
exerted  upon  it  proportional  to  the  instantaneous  current 
squared,  since  the  torque  is  proportional  to  the  product  of  the 
magnetic  field  due  to  the  current  in  the  coil  and  the  field  in- 
duced in  the  iron  vane,  which  is  in  turn  proportional  to  the 
current  if  the  intensity  of  magnetization  is  low.  Hysteresis 
will  not  materially  affect  the  readings  in  connection  with 
alternating  currents  if  the  vane  is  of  very  small  volume.  Such 
instruments  cannot  be  used  readily  as  wattmeters,  and  they  are 
generally  looked  upon  as  less  reliable  than  electro-dynamometer 

instruments  for 
standard  am- 
peremeters and 
voltmeters. 
They  are  much 
used  for  switch- 
board instru- 
ments which 
may  be  expected 
to  be  subjected 
to  currents  of  a 
single  fixed  fre- 
quenc)'.  An  in- 
strument of  this 
type  is  illus- 

Fig.  58. — Plan  of  Thomson  Alternating-current  Amperemeter.  ^ p;) led  with  its 

cover  taken  off  so  as  to  expose  the  working  parts,  in  Fig.  58. 

* Art.  99. 


ELEMENTARY  STATEMENTS 


75 


The  parts  of  this  instrument  are  indicated  by  the  letters,  where 
j D is  the  current  coil,  C the  thin  movable  iron  vane,  B the 
needle,  S the  scale,  and  AA  the  binding  posts  which  are  con- 
nected to  the  coil  I)  by  the  wires  WW. 

e.  Induction  Instruments.  — Induction  instruments  are  made 
on  the  same  principle  as  the  integrating  or  recording  instru- 
ments fully  described  later.  In  the  usual  form,  they  include 
coils  which  induce  currents  in  other  coils  by  transformer  action. 
The  magnetic  fluxes  due  to  the  currents  of  the  various  coils 


WATTMETER 


CURRENT  COIL-5, 


combine  to  form  what  is  called  a rotating  field.  This  in  turn 
sets  up  eddy  currents  in  a movable  disk,  or  shell  of  metal,  to 
which  the  pointer  is  attached,  which  cause  a torque  between 
the  disk  and  coils. 

25.  Arrangement  of  Instruments  for  Measuring  High  Voltages 
or  Large  Currents.  — In  using  a wattmeter  where  very  high 
voltages  are  met,  considerable  difficulty  is  found  in  arranging 
a satisfactory  non-inductive  resistance  for  the  voltage  coil. 
This  difficulty  may  be  over- 
come by  the  use  of  a trans- 
former. Instead  of  connect- 
ing the  voltage  coil  of  the 
wattmeter  across  the  termi- 
nals of  the  test  circuit,  the 
primary  of  a transformer  is 
so  connected,  and  the  vol- 
tage coil  of  the  wattmeter 
is  connected  to  the  sec- 
ondary of  the  transformer  (Fig.  59).  The  constant  of  the 
wattmeter  is  then  dependent  upon  the  ratio  of  transformation 
of  the  transformer,  which  may  be  readily  measured.  This 
method  gives  fairly  reliable  results,  since  the  phases  of  the 
primary  and  secondary  voltages  of  a very  slightly  loaded  trans- 
former are  almost  exactly  180°  apart.  If  the  wattmeter  con- 
stant is  determined  without  the  transformer,  its  constant  when 
in  use  with  the  transformer  must  be  multiplied  by  the  ratio  of 
transformation;  but  wherever  practicable,  the  calibration  ought 
to  be  made  of  the  wattmeter  and  transformer  as  a unit.  In 
like  manner  if  the  current  is  too  great  to  conveniently  pass 
through  the  instrument,  a series  transformer  may  be  used,  as 
shown  in  Fig.  60.  In  such  a case  if  the  wattmeter  has  a scale 


Fig.  59.  — Wattmeter  connected  to  High-vol- 
tage Circuit  through  a Voltage  Transformer. 


7G 


ALTERNATING  CURRENTS 


■WATTMETER 


Fig.  60.  - 


"“ITSTr*- 

SERIES  TRANSFORMER 

-Wattmeter  connected  to  a Circuit  of  Large  Cur- 
rents through  a Series  Transformer. 


for  use  without  the  transformer,  the  readings  must  be  multi- 
plied by  the  ratio  of  transformation  (the  ratio  of  the  main  cur- 
rent to  that  in 
the  transformer 
secondary). 
Changes  of  fre- 
quency change 
the  reading 
slightly  where  a 
series  transform- 
er is  used,  so  that 
it  is  necessary  to 
calibrate  the 
wattmeter  with 
the  transformer 
for  the  frequency  of  the  circuit  to  be  measured.  Evidently 
amperemeters  and  voltmeters  can  have  the  current  or  voltage 
reduced  in  the  same  manner.  For  high-voltage  switchboard 
instruments  it  is  very  desirable  to  use  both  series  current  and 
voltage  transformers  with  the  switchboard  wattmeters,  and 
indeed  with  all  such  switchboard  instruments,  as  in  this  way 
the  danger  of  accident  from  the  high  voltage  can  he  greatly 
reduced. 

Non-inductive  resistance  coils  can  be  used  in  series  with  the 
voltage  coils  of  any  instrument,  as  already  described,  hut  it  is 
quite  difficult  to  secure  a satisfactory  resistance  device  without 
appreciable  self-inductance  or  capacity  for  voltages  larger  than 
a few  hundreds  of  volts.  Shunts  may  be  provided  for  the  cur- 
rent coils,  but  in  the  case  of  instruments  of  the  electro-dynamic 
or  magnetic  vane  type  the  shunts  are  apt  to  cause  frequency 
errors,  and  in  any  event  the  dangers  due  to  direct  connec- 
tion with  the  high-voltage  circuits  are  not  eliminated  as  when 
transformers  are  used. 


CHAPTER  III 


ARMATURE  AND  FIELD  WINDINGS  FOR  ALTERNATORS. 

MATERIALS  OF  CONSTRUCTION. 

26.  Classification  of  Armatures. — Dynamo  armatures  may 
be  classified  under  three  divisions  : 

1.  Where  a wire  cuts  lines  of  force  by  moving  across  them, 

as  in  the  case  of  a slider  or  of  a wire  moving  around  a 

magnet  pole. 

2.  Where  a coil,  or  set  of  coils,  is  moved  parallel  to  itself,  or 

nearly  so,  between  points  of  different  field  strength. 

3.  Where  a coil,  or  set  of  coils,  is  wound  on  a ring  or  drum 

and  given  a rotary  motion  in  a fixed  magnetic  field. 

The  first  division  includes  only  the  so-called  unipolar  arma- 
tures, and  requires  no  treatment  here.  Alternator  armatures 
are  in  general  classed  in  the  second  division. 

It  is  most  usual  for  the  field  to  move  instead  of  the  armature, 
and  in  some  cases  neither  the  armature  nor  field  coils  move,  but 
the  magnetism  linking  the  former  is  varied  by  the  revolution  of 
iron  Inductors.  Where  the  field  rotates,  or  the  variation  of  the 
magnetism  is  effected  by  moving  inductors,  the  voltage  gen- 
erated in  the  armature  windings  is  produced  in  exactly  the 
same  manner  as  if  they  moved,  and  such  armatures  therefore 
belong  to  the  second  division.  That  is,  the  generation  of  the 
voltage  is  dependent  upon  the  relative  motion  of  armature  with 
respect  to  field,  and  this  relative  motion  may  be  effected  by  the 
mechanical  rotation  of  either. 

The  construction  of  the  machines  thus  enumerated  requires 
an  additional  classification  into  : 

1.  Alternators  with  moving  fields. 

2.  Alternators  with  moving  armatures. 

3.  Inductor  alternators. 

The  armatures  of  alternators  belonging  to  the  first  and  sec- 
ond of  these  classes  are  almost  always,  in  the  United  States, 

’ 77 


78 


ALTERNATING  CURRENTS 


Chord  wound,  i.e.  the  winding  of  a coil  passes  across  the  face  of 
the  armature  under  one  pole  face,  and  returns  under  the  next 
pole  of  opposite  sign  without  spanning  an  entire  diameter  of 
the  armature.  The  arrangement  is  the  same  whether  the  arma- 
ture revolves  within  the  field  or  the  field  revolves  within  the 
armature.  Such  a disposition  is  termed  Drum  winding,  in  con- 
tradistinction to  Ring  winding,  where  the  wires  pass  through 
and  around  a ring-shaped  core,  as  in  the  armature  invented  by 
Gramme. 

The  windings  may  further  be  distinguished  as  Bar  and  Coil 
windings.  In  the  former,  rectangular  bars  are  laid  in  slots 
across  the  face  of  the  armature,  and  these  are  properly  con- 
nected together  at  their  ends  by  suitable  welded,  brazed,  or 
bolted  connections ; and  in  the  latter,  coils  of  either  small  rec- 
tangular or  circular  wire  are  wound  upon  a former,  insulated, 
and  then  placed  in  the  armature  slots. 

The  wires  of  a coil  are  sometimes  bunched  together  in  a 


single  slot,  as  shown  in  Fig.  34,  but  are  more  commonly  divided 
into  several  small  coils,  each  occupying  its  own  slot.  The  lat- 
ter arrangement  is  called  Distributed  winding,  because  the  active 
conductors  are  distributed  more  or  less  evenly  over  the  face  of 
the  armature.  (See  Figs.  31,  67,  68,  69,  70.) 

27.  Examples  of  Armature  Windings. — Essentially  the  same 

forms  of  windings  are  used 
for  machines  with  revolving 
armatures  and  machines  with 
revolving  fields,  and  those 
described  in  the  following 
paragraphs  are  applicable  to 
botli  classes.  In  some  of  the 
illustrations,  collector  rings 
and  an  external  crown  of 
poles  are  shown,  for  conven- 
ience, for  revolving  arma- 
tures ; but  the  same  type  of 
winding  may  as  well  be  used 
for  a stationary  armature,  in 
which  case  the  crown  of 
poles  is  internal  and  the  armature  terminals  are  stationary.  The 
original  type  of  armature  winding  used  in  America  is  like  that 


Fig.  Cl.  — Coil  Winding  having  as  many 
Coils  as  Poles. 


ARMATURE  AND  FIELD  WINDINGS  FOR  ALTERNATORS  79 


shown  in  Fig.  61.  In  this  case  the  field  magnet  comprises  a 
ring  of  poles  of  alternate  polarity,  and  the  armature  consists  of 
as  many  coils  A laid  on  the  face  of  the  armature  core  as  there 
are  poles  in  the  field  magnet.  The  ends  of  the  set  of  coils  are 
connected  to  two  collector  rings  It , on  which  bear  brushes  B, 
by  means  of  which  the  current  reaches  the  external  circuit. 
In  this  figure  the  coils  are  shown  for  clearness  as  though  lying 
in  the  plane  of  the  page,  but  in  reality  they  lie  upon  the  face  of 
the  cylindrical  armature.  If  another  set  of  coils  is  placed  90 
electrical  degrees  from  those  shown,  and  connected  to  addi- 
tional collector  rings,  the  winding  is  of  the  two-phase  variety  * ; 
while  if  three  sets  of  coils  are  placed  so  that  the  centers  of  the 
coils  are  60  electrical  degrees  apart,  the  winding  is  three- 
phase.*  In  the  latter  case,  the  coils  are  almost  invariably  con- 
nected to  three  collector  rings,  and  the  connections  must  be 
made  in  the  manner  described  in  Art.  22.  Referring  again  to 
Fig.  61,  it  is  seen  that  adjacent  coils  move  at  any  instant  under 
magnet  poles  of  opposite 
signs,  and  that  the  voltages 
developed  in  them  are  in  op- 
posite directions.  They  must 
therefore  be  connected  in 
circuit  with  each  other  so 
that  they  are  alternately 
right  and  left  handed.  Like- 
wise, when  an  armature 
winding  is  composed  of  a 
number  of  coils  equal  to  one 
half  the  number  of  poles,  as 
illustrated  in  Fig.  62,  the 
coils  must  all  be  connected 
in  the  same  direction  in  the 
circuit. 

To  provide  a winding  similar  to  Fig.  62  for  two  or  three 
phases  involves  adding  the  additional  one  or  two  independent 
sets  of  windings  properly  spaced  and  suitably  connected  to 
appropriate  collector  rings,  as  already  explained  for  the  winding 
of  Fig.  61. 

When  the  coil  connections  cross  each  other  in  their  progress 

* Art.  22. 


Fig.  62.  — Coil  Winding  having  half  as 
many  Coils  as  Poles. 


80 


ALTERNATING  CURRENTS 


around  the  armature,  as  in  Fig.  61,  or  one  half  of  Fig.  63  a,  the 
windings  may  be  called,  after  E.  Arnold*  and  S.  P.  Thompson,! 

Lap  windings.^ 

Alternator  armatures  are  frequently  connected  with  the  two 
halves  of  the  windings  in  parallel,  or,  where  the  machine  is  to 

work  at  low  voltage  and  de- 
liver heavy  currents,  there 
may  be  several  circuits  in  par- 
allel in  each  phase.  In  this 
case,  instead  of  being  ar- 
ranged as  illustrated  in  Fig. 
61,  the  coils  must  be  con- 
nected in  each  half  so  that 
they  are  alternately  right- 
handed  and  left-handed ; and 
where  the  coils  join  the  col- 
lector rings,  both  must  have 
the  same  polarity.  This  is 
indicated  in  Figs.  63  and 
63  a,  which  are  diagrams  for 
Such  a winding,  having  two  circuits 


Fig.  63.  — Coil  Windings  like  Fig.  61,  but 
with  Halves  connected  in  Parallel. 


single-phase  machines, 
in  each  phase,  is  called  a 
Two-circuit  winding.  When 
all  the  coils  are  connected  in 
series,  the  first  and  last  coils 
lie  side  by  side;  consequently, 
in  armatures  built  for  high 
voltages,  a severe  strain  is 
thrown  upon  the  insulation 
separating  them,  and  the 
strain  is  especially  trouble- 
some if  parts  which  differ 
largely  in  potential  occupy 
the  same  slot.  When  the 
halves  of  the  armature  are 
connected  in  parallel,  the 


Fig.  63  a.  — Coil  Winding  like  Fig.  62,  but 
with  Halves  connected  in  Parallel. 


* Die  Ankerioickelungen  der  Gleichstrom-Dynamomachinen . p.  13. 
t Dynamo  Electric  Machinery,  7th  ed.,  vol.  2,  p.  168. 

| For  numerous  diagrams  of  alternator  windings,  see  Parshall  and  Hobart, 
Armature  Windings , Chap.  XII. 


ARMATURE  AND  FIELD  WINDINGS  FOR  ALTERNATORS  81 


4- 


n 


first  and  last  coils  in  the  armature  circuit  lie  on  the  opposite 
sides  of  the  armature,  and  effective  insulation  is  therefore  less 
difficult  to  maintain.  In  the  latter  case  the  voltage  between 
the  adjacent  coils  cannot  be  greater  than  the  total  machine 
voltage  divided  by  half  the  number  of  coils.  When  the  coils 
are  all  in  series,  the  voltage  between  adjacent  coils,  except  the 
two  end  coils,  is 
equal  to  the  total 
voltage  divided  by 
the  total  number  of 
coils.  When  a coil 
winding  of  this  type 
is  connected  with  its 
halves  in  parallel, 
precautions  must  be 
taken  to  make  the 
two  halves  of  the 
winding  alike,  and 
have  the  field  poles 
of  equal  strength ; 
as,  otherwise,  the  in- 
stantaneous voltage 
of  one  half  of  the 
winding  is  likely  to 
differ  from  that  of 
the  other  half,  and 
injurious  cross  cur- 
rents may  occur  in 
the  winding.  This 
precaution,  of 
course,  applies 
wherever  there  are 
several  parallel  cir- 
cuits per  phase. 

Those  forms  of  windings  in  which  the  end  connections  of  any 
one  phase  need  not  cross  each  other  at  the  ends  of  the  armatures 
are  called  by  S.  P.  Thompson*  Wave  windings,  or  according  to 
more  recent  and  acceptable  nomenclature,  Progressive  windings. 
Figures  64  and  65  illustrate  three-phase  progressive  windings. 

* Thompson’s  Dynamo  Electric  Machinery , 7th  ed.,  vol.  2,  p.  168. 


Fig.  64. 


-Three-phase  Progressive  Windings  with 
A Connection. 


82 


ALTERNATING  CURRENTS 


The  first  is  connected  in  A and  the  second  in  Y.  Figure  66 
shows  a winding  of  the  same  type  for  two  phases.  In  these 
figures,  with  the  exception  of  the  developed  views  at  the  top  of 
Figs.  64  and  65,  the  active  conductors  lying  in  the  face  of  the 
armature  are  shown  as  radial  lines,  while  the  end  connections 

are  indicated  by  the 
diagonal  lines  con- 
necting the  active 
conductors  together 
in  such  a way  that 
the  electro-motive 
forces  of  each  circuit 
add  together.  Each 
line  may  represent 
a number  of  wires  in 
the  same  slot,  which 
may  be  arranged  as 
coils  in  each  phase 
in  the  manner  indi- 
cated in  Fig.  62. 

The  principles  of 
armature  windings 
used  in  direct-cur- 
rent machines  are,  in 
general,  applicable 
to  alternators,  but 
practical  considera- 
tions make  it  often 
advisable  to  modify 
the  windings  for  the 


Fig.  65.  — Three-phase  Progressive  Windings  with 
Y Connection. 


latter  machines. 
Synchronous  alter- 
nator armatures  often  have  One-circuit  windings ; that  is,  the 
conductors  of  each  phase  are  connected  in  series,  making  one 
circuit  per  phase,  inasmuch  as  high  voltage  is  usuall)  desit ed. 
Direct-current  armatures,  on  the  other  hand,  have  necessarily 
two-circuit  or  multiple-circuit  windings;  hence,  while  diiect- 
current  windings  may  be  adapted  to  alternating-current  work, 
the  reverse  is  not  always  true. 

The  possibility  of  converting  a direct-current  dynamo  into  a 


ARMATURE  AND  FIELD  WINDINGS  FOR  ALTERNATORS  83 


double-current  machine  has  already  been  referred  to  in  Art.  20. 
A machine  so  constructed,  with  a direct-current  commutator 
and  alternating-current  collector  rings,  may  he  used  to  convert 
a direct-current  which 
is  fed  into  its  commu- 
tator end,  and  by  which 
it  is  driven,  into  an  al- 
ternating-current which 
is  taken  from  the  collec- 
tor rings.  Or,  the  con- 
version may  he  from 
alternating  to  direct 
currents,  if  the  arma- 
ture is  properly  syn- 
chronized so  that  it 
runs  as  a synchronous 
motor.  Or,  alternating- 
current  and  direct-cur- 
rent  may  he  simulta- 
neously produced  or 
utilized  in  such  machines,  or  either  one  of  them  alone.  When 
these  double-current  machines  are  used  for  converting  one 
kind  of  current  into  the  other,  they  are  called  Rotary  convert- 
ers. The  simple  diagram,  Fig.  66  a,  shows  the  principle 

clearly,  though  such  ma- 
chines are  ordinarily  mul- 
tipolar. (For  a discussion 
of  such  machines  see 
Chapter  XII.) 

It  is  possible  in  this 
manner  to  make  a two- 
phaser  to  be  used  with 
separate  circuits  out  of  any 
direct-current  machine 
with  Gramme  or  Siemens 
armature,  by  arranging 
four  collector  rings  on  the 
shaft  and  connecting  them 
to  the  armature  windings 
at  points  which  are  90  electrical  degrees  apart.  It  is  also  possi- 


Fig.  66.  — Two-phase  Progressive  Winding  con- 
nected to  Four  Collector  Rings. 


Fig.  66  a.  — Double-current  Dynamo. 


84 


ALTERNATING  CURRENTS 

ble  to  make  a three -pliaser  out  of  a direct-current  machine  by 
arranging  three  collector  rings  on  the  shaft,  and  connecting  them 
to  the  armature  winding,  at  120  electrical  degrees  apart ; and, 
in  general,  a polyphaser  of  any  number  of  phases  may  be  made 
out  of  a direct-current  machine  by  providing  a proper  number  of 
collector  rings  and  connecting  the  rings  to  appropriate  points  on 
the  armature  winding.  The  taps  for  the  different  rings  should 

360 

connect  with  the  armature  windings  at  points  electrical 

n 

degrees  apart,  where  n is  tire  number  of  phases,  except  that,  for 
single  and  two-phases,  n must  be  taken  equal  to  two  and  four 
respectively.  Suck  machines  may  be  used  to  transform  direct 
currents  into  polyphase  currents  or  vice  versa.  (See  Fig.  66  a.) 

In  connecting  the  collector  rings  of  such  machines  to  the 
armature  windings  the  relative  angles  corresponding  to  the  cur- 
rent phases  must  be  carefully  distinguished.  One  complete 
revolution  of  an  armature  in  a two-pole  field  corresponds  to 
one  complete  period  of  the  alternating  current,  and  therefore 
360  mechanical  degrees  correspond  to  360  electrical  degrees  in 
that  instance,  but  in  multipolar  machines  a rotation  of  the 
armature  equal  to  twice  the  angular  pitch  of  the  poles  corre- 
sponds to  one  complete  period,  so  that,  in  general,  the  relation 

V 

of  electrical  degrees  to  mechanical  degrees  is  ~ : 1,  where  p is 

the  number  of  poles.  Two-pole  alternators  of  the  form  here 
described  evidently  utilize  the  whole  of  the  armature  winding 
with  each  collector  ring  connected  to  a single  point,  as  the 
winding  has  only  two  paths  for  the  current  from  commutator 
brush  to  commutator  brush,  and  the  same  is  true  of  multipolar 
machines  with  two  path  windings.  If  single  connections  to 
the  collector  rings  are  used  in  multipolar  machines  with  multi- 
ple path  windings,  a proportion  of  the  armature  equivalent  to 
360  electrical  degrees  only  is  occupied  in  the  delivery  of  alter- 
nating currents,  and  the  armature  capacity  is  therefore  not 
fully  utilized.  To  fully  utilize  the  armature  in  this  case,  each 
collector  ring  must  be  connected  to  the  winding  at  as  manv 
points  as  there  are  pairs  of  paths,  the  points  being  360  electrical 
degrees  apart ; but  the  different  collector  rings  must  connect 

Q0| ! 

successively  into  points  on  the  armature  winding  which  are  - — 1 
electrical  degrees  apart,  as  already  stated. 


ARMATURE  AND  EIELD  WINDINGS  FOR  ALTERNATORS  85 


In  general,  any  form  of  winding  used  for  direct-current  ma- 
chines,  in  which  alternating  voltages  are  induced  in  the  coils, 
may  be  used  for  alternating-current  machines  by  properly  pro- 
viding collector  rings.  The  only  forms  of  windings  excluded 
from  this  are  the  so-called  acyclic  or  homopolar  machines, 
wherein  the  induced  electro-motive  forces  are  unidirectional. 

28.  Distributed  Windings. — At  the  present  day  distributed 
windings  are  used  almost  universally,  as  they  have  a tendency  to 
reduce  the  armature  reactions,  and  also  may  be  more  readily 
arranged  to  create  a curve  of  voltage  which  closely  approxi- 
mates the  sinusoidal  form.*  The  simplest  form  of  distributed 
winding  is  of  the  coil  type,  distributed  as  in  Fig.  67,  which 


Fig.  67.  — Distributed  Single-phase  Progressive  Coil  Winding. 


shows  three  partial  coils  per  pole,  or  three  slots  per  pole. 
Each  coil  division  may  he  wound  complete  on  a former.  The 
formed  divisions  may  then  be  placed  in  the  slots  and  correctly 
connected  together  to  make  up  the  armature  circuit.  Wind- 
ings of  this  type  can  be  made  for  two  or  three  phases  by  super- 
imposing one  or  two  more  circuits  at  90  or  120  electrical 
degrees  apart  respectively.  For  single-phase  windings  the 


86 


ALTERNATING  CURRENTS 


distances  AB  and  CD  in  the  figure  are  each  usually  made 
nearly  equal  to  BC  to  reduce  differential  action.* 

A common  winding  for  polyphasers  is  a distributed  winding 
of  the  progressive  type  and  is  similar  to  the  two-circuit  chord  wind- 
ings such  as  are  used  on  multipolar  continuous-current  machines. 
This  arrangement  is  sometimes  called  “ barrel  winding.”  Figure 
68  shows  such  a winding  for  six  poles  having  the  Y connection. 
In  this  particular  armature  there  should  be  either  three  or  six 
slots  per  pole  per  phase,  depending  upon  whether  one  or  two 


Fig.  68.  — Three-phase  Winding  with  Y Connection  and  Three  Slots  per  Phase  per 
Pole  in  a Six-pole  Machine. 

sets  of  conductors  are  placed  in  a slot ; and  all  conductors  in  a 
phase  are  connected  in  series,  thus  making  it  a one-circuit  wind- 
ing. It  is  noticed  that,  in  Fig.  68,  after  the  winding  of  one 
phase  has  passed  around  the  armature  three  times  it  doubles 
back  upon  itself  and  passes  around  the  armature  three  times 

* Art.  20. 


ARMATURE  AND  FIELD  WINDINGS  FOR  ALTERNATORS  87 


again.  This  is  to  permit  connecting  the  armature  with  its 
halves  in  parallel  if  desired.  If  the  terminals  of  the  coils,  in- 
stead of  passing  to  collector  rings  at  BI,  BII , and  Bill , re- 
spectively, are  properly  connected  into  the  windings  to  make 
them  reentrant,  and  the  junctions  of  the  interior  connection 
wires  are  connected  to  commutator  bars,  the  winding  is  suitable 
for  a direct-current  machine.  In  fact  the  alternator  windings 
of  the  present  day  are  frequently  the  same  as  those  used  for 
multipolar  direct-current  machinery.  In  tracing  the  winding 


Fig.  69. -Three-phase,  Two-circuit  Winding  with  Y Connection  for  Six-pole 

Machine. 


of  one  phase  through  its  length  it  is  necessary  to  follow  around 
the  armature  six  times  for  the  winding  illustrated  in  Fig.  68, 
which  is  for  a six-pole  machine,  and  the  conductors  in  each 
complete  path  traced  in  going  once  around  the  armature  gen- 
erate a voltage  wave  which  differs  in  phase  from  the  voltage 
wave  of  the  next  set  of  conductors  in  the  same  circuit  by  an 
angle  equal  to  the  electrical  angular  distance  between  the 


88 


ALTERNATING  CURRENTS 


centers  of  adjacent  slots.  As  the  six  slots  in  each  group  of 
Fig.  68  span  60  electrical  degrees,  the  value  of  K will  he  1.055  * 
approximately.  This  same  winding  can  be  used  for  a two-phase 
machine,  when,  instead  of  having  three  sets  of  windings  using 
60°  of  width  each,  the  connections  are  so  made  that  there  are 
two  sets  using  90°  of  width  each.  In  like  manner  a single- 
phase machine  can  be  constructed  by  removing  one  of  the  two- 
phase  windings. 


CONDUCTOR  IN  TOP  OF  SLOT 
CONDUCTOR  IN  BOTTOM  OF  SLOT 


Fig.  70.  — Distributed  Winding  with  Four  Slots  per  Pole  per  Phase  connected  in 
either  Three-phase  A or  Three-phase  Y Connection,  for  a Four-pole  Machine. 

Figure  69  illustrates  the  same  armature  as  Fig.  68,  with  the 
halves  of  the  winding  of  each  of  the  three  phases  connected  in 
parallel.  In  this  figure,  starting  from  any  one  of  the  three 
terminals  BI,  BII , or  Bill , there  are  two  paths  to  follow. 
Each  of  these  paths  passes  entirely  around  the  armature  three 
times  and  occupies  three  slots.  In  the  case  of  a three-phase 
winding,  as  shown,  the  six  slots  of  a phase  should  occupy  one 
third  of  the  polar  pitch  on  the  armature  circumference. 

Figure  70  shows  a distributed  winding  to  occupy  four  slots 
per  pole  per  phase  and  to  be  connected  in  either  three-phase  A 
or  three-phase  Y style,  intended  for  a four-pole  machine.  In 
this  figure  the  winding  is  shown  in  the  developed  form.  i.e.  the 
armature  is  supposed  to  be  rolled  out  upon  a flat  surface. 

* Art.  20. 


ARMATURE  AND  FIELD  WINDINGS  FOR  ALTERNATORS  89 

The  connections  at  the  points  a , b , c are  brought  together  in 
the  finished  winding.  The  terminals  A,  B,  (7,  and  A',  B\  C 
are  the  terminals  which  pass  to  collector  rings,  as  shown  by  the 
A and  Y diagrams. 

Figure  70  a shows  another  form  of  distributed  three-phase 


Fig.  TO  a. — Distributed  Winding  with  Four  Slots  per  Pole  per  Phase,  connected  in 
either  Three-phase  A or  Three-phase  Y Connection,  for  a Four-pole  Machine. 


winding  with  four  slots  per  phase  per  pole,  which  may  be  con- 
nected in  either  A or  Y style.  This  is  a distributed  winding  of 
the  type  shown  for  one  phase  in  Fig.  62. 

The  advantage  of  distributing  the  winding  in  several  slots, 
as  pointed  out  in  Art.  20,  is  the  reduction  of  armature  reaction 
and  more  particularly  the  closer  approximation  toward  a 
sinusoidal  wave  form;  but  the  larger  number  of  conductors 
used  to  produce  a given  effective  voltage  results  in  increased 
self-induction  in  the  armature.  The  question  of  the  number 
of  slots  per  phase  is  settled  after  balancing  the  influences  of 
these  several  effects  and  the  conditions  of  manufacture.  Prac- 
tice indicates  that  three  or  four  slots  per  pole  per  phase  form  a 
satisfactory  compromise,  though  cost  of  insulation  and  manu- 
facture often  make  it  desirable  to  reduce  this  number  at  the 
expense  of  the  wave  form.  When  a sine  wave  form  is  of  prime 


90 


ALTERNATING  CURRENTS 


importance,  a larger  number  than  four  slots  per  pole  per  phase 
is  sometimes  used. 

29.  Disk  Armatures.  — The  disk  form  of  armature  for  alter- 
nators is  one  of  the  earliest  that  came  into  service.  The  first 
commercial  alternator  was  a magneto  machine  (i.e.  with  per- 
manent field  magnets)  known  as  the  Alliance  Dynamo.  This 
was  originally  devised  as  early  as  1849,  but  was  not  developed 
into  commercial  form  until  after  1860.  It  had  a ring  armature. 

In  1867  Wilde  built  an  alternator  with  electro-magnets  and  a 


Fig.  71.  — Disk  Armature  with  Coil 
Winding. 


disk  armature.  From  that  time  on,  the  disk  armature  received 

much  attention  in  Europe  and 
was  an  element  of  many  success- 
ful machines  designed  by  such 
eminent  designers  as  Siemens, 
Ferranti,  and  Mordey.  It  has 
received  less  attention  in  Amer- 
ica, and  no  machines  of  impor- 
tance are  at  the  present  day 
being  actually  manufactured 
with  disk  armatures  on  either 
side  of  the  Atlantic.  This  may 
be  due  to  the'essential  peculiarity 
of  the  disk  which  admits  of  no 
substantial  iron  core  and  is  there- 
fore difficult  to  build  in  a solid  and  workmanlike  manner.  Also, 
it  is  probably  not  possible  to  build 
machines  with  disk  armatures  as 
economically  as  those  in  which  the 
armatures  are  built  upon  substan- 
tial iron  cores;  at  least  when  the 
two  types  are  made  of  equal  me- 
chanical stability.  Disk  armatures 
may  be  wound  either  with  lap  or 
progressive  windings.  Figure  71 
illustrates  three  coils  of  a coil- 
winding arranged  to  rotate  be- 
tween poles  of  opposite  polarities,  Fig.  72.  — Disk  Armature  with 

i ,,  , r Progressive  Winding, 

and  a section  through  one  pair  ot  ° 

poles.  Figure  72  shows  a progressive  winding.  Either  the 

armature  or  the  field  may  revolve.  (Examples:  Ferranti 


ARMATURE  AND  FIELD  WINDINGS  FOR  ALTERNATORS  91 


Alternator;  Mordey  Alternators;  Brush  Alternator.)  The 
absence  of  iron  in  disk  armatures  reduces  the  losses  due  to 
hysteresis  and  eddy  currents  to  a minimum,  but  it  increases 
the  depth  of  the  air  gap.  Hence,  a greater  exciting  current 
is  apt  to  be  required  for  the  field  magnets,  and  many  turns 
must  be  placed  upon  the  armature.  That  this  objection  may 
be  overcome  is  shown  by  the  small  amount  of  energy  required 
to  magnetize  the  old  Mordey  alternators.  In  a 75-kilowatt 
machine  of  this  type  the  I2R  loss  in  the  armature  is  2.3  per 
cent  and  in  the  field  is  1.5  per  cent,  which  compare  favorably 
with  the  losses  in  other  machines.  The  curve  of  voltage  in 
disk  armatures  is  in  general  quite  near  to  that  of  the  sine  func- 
tion. The  first  experimental  determination  of  the  form  of  the 
curve  was  made  in  1880  by  Joubert,  who  experimented  upon  a 
Siemens  machine  having  a disk  armature.  The  curve  proved 
to  be  practically  a sinusoid.  This  is  also  true  of  the  curve  of 
voltage  developed  by  the  Mordey  alternator. 

A diagram  of  a Mordey  alternator  is  shown  in  Fig.  73.  It 
is  seen  that  in  the  construction  of  this  machine  only  a single 
exciting  coil  F is  required,  and  thus  the 
expense  of  construction  is  in  some  de- 
gree reduced.  In  this  machine  all  poles 
of  the  same  sign  lie  upon  the  same  side 
of  the  armature.  The  armature  coils  are 
arranged  in  a disk,  which  is  stationary 
between  the  faces  of  the  revolving  poles. 

The  Ferranti  alternator  has  two  crowns 
of  pole  pieces  and  magnet  coils  at  either 
side  of  the  disk,  as  shown  in  Fig.  71. 

The  poles  in  this  machine  which  lie  on 
one  side  of  the  armature  alternate  in  po- 
larity. The  Ferranti  alternator  is  of 
especial  interest  on  account  of  the  Fer- 
ranti machines  installed  at  Deptford 
Station  near  London,  which  was  the  first 
high-voltage,  long-distance  transmission  plant  of  any  impor- 
tance. The  Ferranti  and  Mordey  machines  are  now  substan- 
tially obsolete. 

In  iron-cored  machines,  armature  reactions  have  a more  dis- 
torting effect  and  therefore  tend  to  modify  the  form  of  the  vol* 


Fig.  73.  — Diagram  of  a 
Mordey  Alternator. 


92 


ALTERNATING  CURRENTS 


Fig.  74.  — Diagram  of  a Revolving 
Ring  Armature. 


tage  curve.  The  variation  is  usually  not  great  in  machines  hav- 
ing distributed  windings,  but  where  single-coil  windings  are 
used  on  single-phasers,  the  curves  of  voltage  may  deviate  wide- 
ly from  the  sinusoid,  and  sometimes  become  quite  irregular. 

30.  Other  Types  of  Armatures.  — Ring-wound  alternator  ar- 
matures were  early  used  with  commercial  success,  and  some  of 
the  older  magneto  machines  of  the  De  Meritens  type  with  per- 
manent field  magnets  and  ring  armatures  are  still  in  existence. 
The  invention  of  the  ring  armature  for  alternating-current 
machines  is  usually  ascribed  to  Gramme  or  Wilde,  who  inde- 
pendently patented  the  form  in 
France  and  England  in  1878. 
In  America  ring  armatures  have 
never  been  viewed  with  as  great 
favor  as  have  drum  armatures, 
and  they  are  not  used,  partially 
on  account  of  the  small  mechani- 
cal stability  of  the  ring  and  the  greater  self-induction  of  its 
windings,  hut  more  especially  on  account  of  the  fact  that  the 
windings  must  be  placed  by  hand  and  are  of  greater  length, 
which  conditions  result  in  excessive  expense.  Figure  74  is  a 
diagram  of  a segment  of  a revolving  ring  armature  intended  to 
revolve  inside  of 
an  inwardly  point- 
ing crown  of  poles 
of  alternate  polar- 
i t i e s . Such 
armatures  can,  of 
course,  be  mechan- 
ically arranged  so 
that  field  magnets 
with  outwardly 
pointing  poles 
may  be  located 
inside  the  ring 
instead  of  the  re- 
verse, and  either  armature  or  fields  may  be  arranged  to  revolve 
in  any  particular  instance. 

In  some  of  the  older  types  of  foreign  alternators  the  arma- 
ture coils  were  wound  upon  long  projections  similar  to  the  field 


Fig.  75.  — Alternator  with  Pole  Armature. 


ARMATURE  AND  FIELD  WINDINGS  FOR  ALTERNATORS  93 


cores,  and  these  were  called  Pole  armatures.  On  account  of 
the  great  variation  in  reluctance  of  the  magnetic  circuit  as  the 
armature  revolves,  such  armatures  cause  large  losses  in  the 
pole  pieces  by  eddy  currents,  unless  the  iron  field  is  entirely 
laminated.  One  form  of  pole  armature  is  shown  in  Fig.  75. 
In  this  machine  the  armature  is  stationary  while  the  field  re- 
volves. (Example:  early  Ganz  alternators  and  others.)  In  the 
form  illustrated  in  the  figure,  the  reluctance  of  the  magnetic 
circuit  varies  less  as  the  poles  move  past  the  armature  coils,  on 
account  of  the  fact  that  the  armature  coils  are  really  wound  in 
deep  slots,  thus  reducing  the  type  to  a close  approximation  of 
the  drum  windings  on  slotted  cores,  as  already  described. 

31.  Methods  of  Applying  the  Wires  to  Chord-wound  Arma- 
tures. — The  coils  for  alternator  armatures  are  ordinarily 
wound  on  formers  and  then  fastened  upon  the  armatures  after 
having  been  well  insulated.  At  present  the  armature  wind- 
ings are  nearly  always  placed  in  slots,  which  are  properly 
formed  in  the  surfaces  of  the  cores,  but  most  of  the  older 
American  machines  had  their  windings  fastened  upon  the  sur- 
faces of  smooth  armature  cores. 

In  the  old  machines  where 
surface  windings  were  used, 
the  coils  were  frequently  ar- 
ranged to  bend  down  over  the 
ends  of  the  cores,  as  illustrated 
in  Fig.  76,  where  they  could 
be  securely  fastened  by  end 
plates  or  blocks  of  wood  or 
fiber.  It  was  usual  to  fill  the 

. , „ Fig.  76. — Surface-wound  Alternator 

spaces  in  the  centers  of  the  Armature. 

coils  with  wood  blocks  screwed 

to  the  cores  or  held  by  binding  wires.  (Examples:  early 
Westinghouse  and  National  alternators.)  In  some  machines 
the  coils  were  flat  or  pancake-like,  and  of  the  same  length  as 
the  armature  cores.  In  this  case  they  were  laid  upon  the 
cylindrical  surfaces  of  the  armature  cores  and  securely  bound 
with  wire  bands.  (Examples:  early  Thomson-Houston  and 
similar  alternators.)  The  high  peripheral  velocities  of  alter- 
nator armatures  make  heavy  bands  essential  to  the  preserva- 
tion of  surface  windings,  and  all  blocking  must  be  securely 


94 


ALTERNATING  CURRENTS 


fastened.  The  wood  blocks  which  fill  the  center  of  the  coils 
make  excellent  driving  teeth,  and  therefore  serve  a good  pur- 
pose if  they  are  fastened  securely  to  the  core. 

When  imbedded  coils  are  used,  they  may  be  made  upon 
formers  (lathe-wound),  and  then  applied  to  the  core,  or  the 
conductors  may  be  threaded  through  the  grooves  which  are  pro- 
vided with  insulating  linings.  When  the  core  teeth  are 
T-shaped,  the  coils  are  sometimes  wound  of  sufficient  width  to 
slip  over  the  head,  and  when  in  place,  they  may  be  narrowed  by 
squeezing.  The  coils  for  this  purpose  must  be  wound  with  the 
ends  of  such  shape  that  they  will  bend  without  injury,  as  is 
shown,  for  example,  in  Fig.  77.  The  methods  used  in  manufac- 
turing the  coils 
and  applying 
them  to  slotted 
armature  cores 
of  direct-cur- 
rent machines 
may  also  be 
used  in  the  con- 

Fig.  77.  — A Method  of  Inserting  Coils  over  T-shaped  Teeth.  . „ 

struction  of  al- 
ternator armatures  with  distributed  windings  ; but  the  alternator 
armatures  are  usually  of  much  higher  voltage,  and  the  insulation 
must  be  carefully  and  specially  designed.  Construction  of  this 
character  (namely,  using  small  distributed  coils)  has  largely 
superseded  construction  of  the  character  illustrated  in  Fig.  77. 

Lathe-wound  coils  are  of  decided  advantage  for  armatures 
designed  for  high  voltages,  as  their  insulation  may  be  made 
particularly  safe.  Such  coils  are  usually  served  with  layers  of 
japanned  canvas,  special  fuller  or  press  board  made  for  insulat- 
ing purposes,  vulcanized  fiber,  and  mica.  The  slots  between 
the  core  teeth  may  be  made  of  sufficient  area  to  permit  the  use 
of  any  desirable  thickness  of  insulation,  and  the  teeth  afford 
very  complete  mechanical  protection  for  the  coils.  Toothed 
armatures  with  lathe-wound  coils  should  therefore  be  thoroughly 
reliable.  If  the  magnetic  surface  of  the  armature  is  fairly 
complete,  the  wires  are  protected  from  magnetic  drag,  which 
decreases  the  chances  of  the  conductors  chafing  and  injuring 
the  insulation.  The  winding  shown  in  Fig.  77  is  not  distrib- 
uted, but  grouped  in  one  slot  per  pole,  as  shown  also  in  the 


ARMATURE  AND  FIELD  WINDINGS  FOR  ALTERNATORS  95 


Fig.  78. — Two-phase  Armature  Construction. 


96 


ALTERNATING  CURRENTS 


diagrams  of  Figs.  61  to  63.  Distributed  windings  have  very 
largely  superseded  this  type,  as  already  explained. 

The  construction  of  a two-phase  group-wound  armature,  hav- 
ing open  rectangular  winding  slots  in  the  core,  is  shown  in 
Fig.  78.  Each  phase  in  this  case  forms  a complete  single-phase 
winding,  as  in  Figs.  61  and  63.  It  is  to  be  noticed  that  the 
forms  upon  which  the  coils  of  one  phase  are  wound  are  of  such 
a shape  that  the  ends  of  the  coils  bend  down  over  the  ends  of  the 
armature,  while  the  coils  of  the  other  phase  are  rectangular. 
This  construction  keeps  the  coils  well  apart  at  points  of  cross- 
ing. After  the  coils  are  properly  insulated  they  are  slipped 
into  the  rectangular  slots,  and  are  held  in  place  by  wedge-shaped 
strips  of  hard  wood.  Grooves  are  arranged  near  the  upper 
edges  of  the  iron  teeth  into  which  the  wooden  strips  fit.  In 
this  manner  the  windings  may  be  held  very  firmly  without  the 
aid  of  band  wires.  The  four  peripheral  slots  to  be  observed 
in  the  core  are  ventilating  openings,  maintained  by  fingered 
grids  inserted  between  the  adjacent  core  stampings. 

The  way  in  which  coils  are  arranged  for  insertion  into  the 
partially  closed  slots  of  the  modern  distributed  armature  wind- 
ings is  shown  in  Fig.  79.  One  end  of  each  coil,  which  in  this 
case  is  composed  of  several  turns  of  round  wire,  is  left  open. 
The  coil  is  then  slipped  through  the  proper  slots,  and  the  several 
wires  are  joined  to  complete  the  coil.  In  this  figure,  each  coil 
occupies  four  slots  and  is  therefore  a distributed  winding. 
The  coil  terminals  whereby  the  coils  are  connected  with  each 
other  are  shown  as  spirally  insulated  wires  at  the  top  of  the 
figure.  Another  and  similar  winding  for  partially  closed  slots 
is  shown  in  Fig.  80.  This  winding,  as  seen,  is  placed  upon  a 
stationary  armature  intended  to  surround  a rotating  field,  and 
copper  straps  or  bars  are  used  for  making  the  coils  instead  of 
round  wire.  To  complete  the  winding,  the  open  ends  of  the 
coils  must  be  brazed  together  and  the  coils  themselves  be  inter- 
connected. Figure  81  shows  how  the  straps  are  brought 
around  ready  for  brazing,  and  also  shows  the  completed  joints 
after  the  final  insulating  wrapping  has  been  applied.  In  the 
left-hand  side  of  the  figure  may  be  seen  the  connectors  be- 
tween coils.  The  method  of  insulating  the  slots  is  shown  in 
the  lower  right-hand  corner  of  the  figure ; the  two  slots,  which 
have  not  as  yet  received  their  coils,  show  the  tubes  of  insulat- 


ARMATURE  AND  FIELD  WINDINGS  FOR  ALTERNATORS  97 


Fig.  79.  — Grouping  of  Wire  Coils  for  Partly  Closed  Slots. 


98 


ALTERNATING  CURRENTS 


J?ig.  80.  — Strap  or  Bar  Coils  in  Partly  Closed  Slots.  Incomplete  Winding. 


ARMATURE  AND  FIELD  WINDINGS  FOR  ALTERNATORS  99 


ing  material  — micanite,  fuller  board,  or  other  insulator  — 
extending  out  a short  distance  from  the  core.  The  method  of 
taping  the  coils  to  complete  the  insulation  can  also  be  readily 
seen.  The  conductors  are  held  firmly  in  place  by  wooden 
wedges  driven  in  the  tops  of  the  slots  in  the  manner  more 
plainly  seen  in  Fig.  78. 

In  Fig.  82  is  shown  a three-phase,  coil-wound,  stationary 
armature  having  a distributed  winding  of  four  slots  to  the  coil. 


Fig.  81.  — Method  of  Connecting  Open  Ends  of  Strap  Coils. 


The  particular  machine  from  which  the  photograph  was  taken 
is  wound  to  generate  a voltage  of  13,000  volts,  and  is  therefore 
very  heavily  insulated. 

Figure  83  shows  a model  of  a three-phase  partially  distrib- 
uted winding  applied  to  one  of  the  5000-kilowatt  generators 


100 


ALTERNATING  CURRENTS 


Fig.  82.  — Three-phase  Winding  for  13,000  Volts. 


ARMATURE  AND  FIELD  WINDINGS  FOR  ALTERNATORS  101 


Fig.  83. — Model  of  Three-phase  Partially  Distributed  AViuding. 


102 


ALTERNATING  CURRENTS 


of  the  Manhattan  Station  in  New  York  City.  It  is  especially 
interesting  as  showing  the  manner  in  which  the  windings  are 
arranged  for  crossing  at  the  ends  and  the  methods  of  construc- 
tion and  insulation. 

The  lap,  progressive,  and  barrel  windings  given  heretofore 
are  suitable  for  any  type  of  generator,  including  those  used 
with  steam  turbines.  Figure  83  a shows  the  stationary  arma- 
ture of  a horizontal  steam  turbine  generator  of  1000-kilowatt 
capacity  of  Westinghouse  type.  It  is  evident  from  the  con- 


Fig.  83  a.  — Turbine  Generator  Armature. 

struction  shown  in  the  figure  that  this  type  of  generator  is  of 
the  same  character  as  those  driven  in  a different  manner.  The 
high  speed  of  such  machines  makes  it  necessary  to  wind  them 
for  a comparatively  small  number  of  poles.  The  winding  in 
the  figure  is  two-phase,  four-pole,  of  the  distributed  coil  type. 

32.  Collectors.  — In  all  the  alternators  heretofore  described, 
either  the  field  or  the  armature  windings  are  arranged  to  re- 
volve. In  the  case  of  inductor  alternators  to  be  described 
later  it  is  possible  to  arrange  the  construction  so  that  both  the 
magnetizing  and  armature  coils  may  remain  stationary,  but 
mechanical  and  other  considerations  ordinarily  render  this  inad- 


ARMATURE  AND  FIELD  WINDINGS  FOR  ALTERNATORS  103 


visable.  It  therefore  may  be  said  generally  that  either  the  arma- 
ture or  the  field  windings  must  revolve  with  the  iron  cores  on 
which  they  are  wound.  This  makes  essential  the  use  of  some 
means  for  conveying  the  current  to  and  from  the  revolving  coils. 

In  either  case  no  modification  of  the  current  is  required  and 
therefore  plain  insulated  Collector  rings  serve  the  purpose  when 
they  are  properly  mounted  on  the  shaft  so  that  the  brushes  may 
be  arranged  to  bear  against  them.  Figure  8-1  is  an  illustration 
of  collector  rings  for  conveying  the  current  to  a multipolar 
revolving  field.  These  rings,  often  called  Collectors,  are  some- 
times made  of  copper  or  bi'onze,  though  they  are  now  com- 
monly made  of  cast  iron.  In  the  older  construction  the  rings 
were  imbedded  in  cylinders  of  vulcanized  fiber  or  the  like 
keyed  to  the  shaft  of  the  rotating  part.  In  modern  construc- 
tion the  rings  are  usually  placed  upon  a metal  spider  and  hub. 
This  makes  it  possible  to  do  away  with  such  a large  bulk  of 
insulation,  and  thus  adds  materially  to  the  mechanical  rigidity 
of  the  construction.  The  insulating  materials  used  are  ordi- 
nary pressed  fiber,  vulcanite,  or  some  form  of  mica  board. 
Figure  92  shows  four  rings  mounted  between  discs  of  solid 
insulation,  for  a four-phase  rotating  armature ; and  Fig.  84 
shows  two  rings  on  metal  spiders  affording  air  insulation. 

Collection  is  obtained  by  means  of  brushes  fastened  to  the 
brush  holders  and  hearing  upon  the  rings.  These  brushes  are 
usually  of  carbon,  electroplated  with  copper  to  increase  their 
conductivity.  The  copper  coating  is  cut  away  from  the  end 
of  the  brush  which  bears  on  the  ring.  As  carbon  has  a high 
resistance  it  is  not  suitable  for  machines  delivering  a very 
large  amount  of  current.  In  such  cases  it  is  common  to  use 
brushes  made  of  fine  woven  copper  wire  gauze.  Former  prac- 
tice employed  brushes  made  of  copper  strips,  but  as  it  is  diffi- 
cult to  make  strip  brushes  which  will  not  cut  the  rings,  the 
substitution  of  carbon  and  woven  wire  has  become  almost 
universal. 

A carbon  brush  can  collect  without  undue  heating  from 
about  40  to  80  amperes  per  square  inch  of  contact  surface, 
while  a woven-wire  brush  can  collect  over  two  or  three  times 
that  amount.  A number  of  brushes  may  make  contact  with 
each  ring. 

The  brushes  are  mounted  in  brush  holders,  which  in  turn  are 


104 


ALTERNATING  CURRENTS 


Fig.  84.  — Rotating  Field  Magnet,  showing  Collector  Rings,  and  also  showing  Exciter  Armature  mounted  on  Same  Shaft. 


ARMATURE  AND  FIELD  WINDINGS  FOR  ALTERNATORS  105 


fastened  to  the  frame  of  the  machine  some  convenient  arms 
or  brackets;  the  circuit  connections  are  often  made  to  the 
brushes  by  means  of  flexible  copper  cords  or  strips  soldered  on, 
in  addition  to  the  contact  of  brush  with  holder.  The  brush 
holders  are  so  arranged  through  the  medium  of  springs  that 
the  brushes  may  be  made  to  bear  upon  the  collector  rings  with 
any  desired  pressure.  Ordinary  practice  calls  for  a pressure  of 
from  1 to  1^  pounds  per  square  inch.  A greater  pressure  is  apt 
to  cause  cutting  of  the  rings  and  a less  pressure  may  permit 
the  brushes  to  vibrate  sufficiently  to  cause  sparking. 

The  safe  limit  for  the  amount  of  current  that  can  be  col- 
lected per  square  inch  is  fixed  by  the  heating  alone  and  there- 
fore may  be  quite  large  without  danger.  If  mechanical 
considerations  required  it,  doubtless  as  much  as  500  amperes 
per  square  inch  of  contact  might  be  satisfactorily  collected 
by  means  of  copper  gauze  brushes  and  as  much  as  200  amperes 
by  carbon  brushes,  but  such  extreme  cases  do  not  often  arise 
and  it  is  not  generally  advisable  to  exceed  one  fourth  these 
amounts. 

Instead  of  brushes  the  collection  may  be  effected  by  means  of 
flexible  weighted  copper  bands  hung  from  the  rings.  Figure  85 
indicates  such  an  arrangement.  An 
arrangement  somewhat  similar  to 
that  of  Fig.  85  was  used  on  the 
Ferranti  alternators  of  the  famous 
Deptford  Station  already  referred 
to.*  By  this  construction  a large 
collecting  area  may  be  gained  with- 
out unduly  increasing  the  size  of 
the  rings,  but  the  arrangement  of 
carbon  brushes  rubbing  on  flat  rings 
is  mechanically  preferable.  As 
modern  alternators  are  usually  of 
the  revolving  field  type  and  the  collectors  are  used  for  trans- 
mitting exciting  current  to  the  revolving  field  coils,  the  collec- 
tors need  not  be  so  highly  insulated  as  would  be  necessary  for 
a high  alternating  voltage.  The  field  voltage  seldom  exceeds 
220  volts.  It  is  good  practice  to  mount  the  two  rings,  for  this 


Fig.  85.  — Copper-band  Contactors. 


* Art.  29. 


106 


ALTERNATING  CURRENTS 


purpose,  upon  separate  spiders  attached  to  one  hub,  so  that  there 
is  an  air  space  between  the  rings. 

33.  Field  Excitation  of  Alternators. — The  exciting  current 
for  synchronous  alternators  must  be  unidirectional,  and  it  is 
ordinarily  obtained  from  auxiliary  direct-current  shunt  or  com- 
pound dynamos  called  Exciters,  as  illustrated  in  Fig.  86.  If 
the  alternator  field  magnets  are  stationary,  this  exciting  current 

is  fed  directly  into  the  coils, 

A RM  ATI  IDP 

but  if  the  machine  is  of  the 
revolving  field  type,  the  ex- 
citing current  must  be  led 
into  the  coils  through  col- 
lector rings ; in  this  case 
the  armature  current  is  led 
directly  from  the  terminals 
of  the  stationary  armature. 

Fig.  86.  — Diagram  of  a Separately  Excited  The  eXciter  is  fluently 
Alternator,  with  Bus  Bars  for  Parallel  Connec-  driven  from  the  same  prime 
tion  of  Exciters  and  Alternators.  Switches,  moygr  as  is  its  alterna- 
etc.,  not  Shown.  . . 

tor,  and  is  sometimes  even 

mounted  upon  the  same  shaft.  In  generating  plants  having 
large  units,  however,  it  is  better  practice  to  have  the  exciters 
driven  by  separate  prime  movers,  in  which  case  two  or  three 
exciters  may  be  provided  to  furnish  the  exciting  current  for  all 
the  main  generators.  The  regulation  of  the  alternator  voltage 
is  secured,  by  varying  the  resistance  in  the  field  circuit  of 
the  alternator  or  by  varying  the  field  strength  of  the  exciter. 
These  are  the  usual  methods  in 
America  and  are  done  both  by 


Fig.  87.  — Self-excited  Alter- 
nator. 


Fig.  88.  — Compositely  Excited  Alter 
nator. 


ARMATURE  AND  FIELD  WINDINGS  FOR  ALTERNATORS  107 


hand  or  by  an  automatic  controller,  as  is  explained  in  detail  in 
the  chapter  dealing  with  alternator  operation.  Self-excitation 
and  composite  ex- 
citation, as  shown 
below,  were  much 
used  earlier  and 
have  sufficient  pos- 
sibilities to  war- 
rent  study. 

By  suitable  com- 
mutation the 
alternator  may 
evidently  be  made 
to  furnish  its  own 
exciting  current 
either  wholly  or  in 
part,  and  the  windings  of  the  field  magnets  of  alternators  may 
be  classified  according  to  their  arrangement  in  circuit.  The 
principal  divisions  are  three  : separately  excited,  self-excited, 
compositely  excited;  so-called,  respectively,  when  the  exciting 
current  is  supplied  from  an  external  source  (Fig.  86),  when  it 
is  supplied  through  a rectifying  commutator  from  the  armature 
of  the  machine  under  consideration  (Fig.  87),  or  when  these 
two  arrangements  are  combined  (Fig.  88).*  Self-excited  al- 
ternators may  again 
be  d*i  v i d e d into 
series-wound  and 
shunt-wound,  de- 
pending upon,  first, 
whether  the  whole 
current  is  rectified 
and  led  through  a 
comparatively  small 
number  of  turns 
around  the  field 
magnets  (Fig.  89), 
or,  second,  whether 
only  a portion  of  the  current  is  rectified  and  led  through  a 

* Compare  Jackson’s  Electromagnetism  and  the  Construction  of  Dynamos, 

p.  136. 


Fig.  90.  — Diagram  of  a Shunt-wound  Alternator. 


Fig.  89.  — Diagram  of  a Series-wound  Alternator. 


108 


ALTERNATING  CURRENTS 


shunt  circuit  many  times  around  the  magnets  (Fig.  90).  In 
the  latter,  either  the  whole  voltage  of  the  armature,  or  that  of 


Fig.  91.— Diagram  of  a Shunt-wound  common,  as  the  inconveniences 


than  outweigh  any  advantages  derived  from  making  the  machines 
self-contained. 

Composite  Excitation.  — Evidently  another  division  might  be 
added  to  those  named  in  the  preceding  paragraph,  which  would 
comprise  a compound  winding  in  which  both  the  shunt  and  series 
field  currents  are  supplied  by  rectification.  This,  however, 
would  require  two  rectifying  commutators,  which  at  the  best 
are  unsatisfactory,  and  for  other  reasons  would  not  prove  prac- 
tical. To  gain  the  result  for  which  compounding  is  used  in 
direct-current  dynamos,  the  composite  winding  is  used.  That 
is,  the  alternator  is  externally  excited  to  its  normal  voltage  on  open 
circuit  and  the  internal  losses  are  compensated  by  series  ampere- 
turns  from  self -excitation.  This  method  of  excitation  is  common 
with  machines  having  rotating  armatures  and  is  very  desirable  in 
small  or  medium-sized  electric  light  plants  or  where  the  nature 
of  the  service  calls  for  constant  regulation  with  meager  switch- 
board attendance.  The  series  turns  may  be  so  arranged  that 
they  will  add  more  than  the  voltage  lost  in  the  machine  itself, 
thus  providing  some  compensation  for  the  voltage  lost  in  the 
feeder  wires  and  tending  to  maintain  the  voltage  at  the  center 
of  distribution  approximately  constant.  Evidently,  in  the  latter 
case  the  voltage  at  the  machine  terminals  will  rise  when  the  load 
increases.  Under  such  circumstances  the  machine  is  said  to  be 
Over-compounded,  and  the  term  Compound-wound  has  of  recent 


one  or  more  coils,  may  be  im- 
pressed upon  the  rectifying 
commutator  either  directly,  as 
illustrated  in  Fig.  90,  or  by 
means  of  a transformer  at- 
tached to  the  armature.  Fig-- 
ure  91  illustrates  the  trans- 
former arrangement  when  the 
fields  rotate.  Self-excited  al- 
ternators are  now  quite  un- 


Re  volving  Field  Alternator,  with  the 
Voltage  reduced  for  Commutation  by  a 
Transformer. 


accompanying  the  rectification 
of  the  current  under  usual  con- 
conditions  of  operation  more 


ARMATURE  AND  FIELD  WINDINGS  EOR  ALTERNATORS  109 


Fig.  92.  — Revolving  Armature  arranged  with  Series  Transformers  for  Composite  Excitation.  A A,  Series  Transformers. 


110 


ALTERNATING  CURRENTS 


years  come  into  general  use  to  designate  these  machines  instead 
of  the  more  distinctive  term  composite-wound. 

The  self-exciting  circuit  of  the  composite  winding  may  be 
arranged  in  various  ways.  Thus  the  armature  current  may  all 
be  rectified  for  use  in  excitation  (Fig.  93)  or  it  may  pass 


Fig.  93. — Diagram  showing  All  of  the  Armature  Current  rectified  for 
Field  Excitation. 

through  special  series  transformers  attached  to  the  armature, 
and  the  secondary  of  these  may  then  supply  the  current  for  rec- 
tification and  self-excitation  (Figs.  92  and  112).  The  cores 
of  these  transformers  may  be  either  independent  of  the  arma- 


Fig.  94.  — Diagram  of  Composite  Excitation  with  Series  Turns  on  All  Poles. 

ture  core,  as  at  A in  Fig.  112,  or  may  consist  of  the  lami- 
nated spider  or  other  portions  of  the  armature  core.  Again, 
the  rectified  current  may  be  passed  through  a few  turns  of  wire 


ARMATURE  AND  FIELD  WINDINGS  FOR  ALTERNATORS  111 


on  each  pole  (Fig.  91),  or  all  the  necessary  series  turns  may  be 
concentrated  upon  one  or  two  poles  (Figs.  95  and  96).  In  the 
latter  case,  the  series  turns  must  always  be  equally  divided 
between  two  poles  with  symmetrical  positions,  as  in  Fig-.  96, 


Fig.  95.  — Diagram  of  Composite  Winding  with  Series  Turns  on  One  Pole. 

when  the  armature  winding  is  connected  with  its  halves  in  par- 
allel. Composite  windings  may  be  arranged  with  the  self-exci- 


Fig.  96.  — Diagram  of  Composite  Winding  with  Series  Turns  on  Two  Poles. 

tation  in  a shunt  circuit,  but  no  advantage  is  gained  by  this 
arrangement  over  complete  self-excitation  in  shunt  or  over  sepa- 
rate excitation,  and  the  arrangement  is  never  used.  In  some 


112 


ALTERNATING  CURRENTS 


old-type  self-exciting  alternators,  a separate  set  of  exciting 
coils  was  wound  on  the  armature  and  connected  to  a rectifying 

commutator.  These  were 
wound  either  directly  with 
the  main  armature  coils,  or 
across  a chord  of  the  arma- 
ture core,  as  illustrated  in 
Fig.  9'  7,  in  which  the  coils  A 
for  generating  exciting  cur- 
rent are  wound  across  a long 
chord  of  the  armature,  as 

Fig.  97.  — Method  of  Winding  Special  shown  in  the  upper  diagram 

Alternator  Exciting  Coil.  „ , . „ . .. 

ot  the  figure,  while  the  mam 
armature  coils  B are  wound  pancake  fashion.  The  compound- 
ing may  be  effected  in  self-excitecl  alternators  by  means  of 
shunt  and  series  transformers  combined  as  in  Fig.  98,  which 
shows  a machine  with  stationary  armature.  This  arrangement 
pertains  to  some  early  European  examples  put  out  by  Ganz  and 
Co.  The  arrangement 
permits  a self-excited 
alternator  to  be  com- 
pounded by  the  use  of 
only  one  commutator. 

The  shunt  transformer 
furnishes  what  is  equiva- 
lent to  the  shunt  current 
of  an  ordinary  com- 
pound dynamo,  and  the 
series  transformer  adds 
additional  voltage  to  increase  the  current  as  required  for  com- 
pounding. This  latter  voltage  increases  with  the  load  on  the 
alternator  as  the  primary  winding  of  the  transformer  is  in 
series  with  one  of  the  alternator  leads.  In  the  illustration,  S 
represents  the  series  transformer,  T the  shunt  transformer,  and 
li  a variable  rheostat  to  control  the  degree  of  excitation. 

For  the  purpose  of  varying  the  magnetizing  effect  of  the 
series  turns,  a variable  shunt  is  often  connected  across  their 
terminals  in  a manner  similar  to  the  usage  with  direct-current 
machines,  as  at  A in  Fig.  99,  and  a shunt  is  sometimes  placed 
across  the  rectifier  terminals  in  such  a way  that  only  a fixed 


Fig.  98.  — Diagram  of  Compounding  by  Means 
of  Shunt  and  Series  Transformers. 


ARMATURE  AND  FIELD  WINDINGS  FOR  ALTERNATORS  113 


proportion  of  the  total  current  is  rectified  and  passes  through 
the  series  field  winding,  as  at  B in  Fig.  99.  This  latter  shunt 
reduces  the  difficul- 
ties caused  by  spark- 
ing by  reducing  the 
current  to  be  rec- 
tified. 

Polyphase  alter- 
nators may  be  com- 
pounded in  the  man- 
ner described  above 
by  rectifying  the 
current  of  one  phase. 

Another  method 
of  compounding  is 
based  upon  the  mag- 
netic reactive  effect  of  current  in  an  armature  upon  the  field. 
The  revolving  field  magnet  of  such  a machine,  called  a “ compen- 
sated alternator,”  is  shown  in  Fig.  100.  Upon  the  same  shaft 
as  the  revolving  field  magnet  is  placed  the  direct-current  ex- 
citer armature  and  its  commutator.  Surrounding  the  revolv- 
ing field  magnet  is  a three-phase  stationary  distributed  armature 
winding  of  the  ordinary  type,  and  supported  by  the  same  frame 

ALTERNATOR  FIELD  RINGS 


Fig.  100. — Revolving  FieldMagnet  and  Exciter  Armature  of  a Compensated  Alternator. 


Fig.  99.  Diagram  of  Composite  Excitation  having 
the  Series  Field  and  the  Rectifier  Shunted. 


is  the  exciter  field,  which  has  the  same  number  of  poles  as  the 
field  of  the  alternator.  Direct  current  is  drawn  from  the  com- 
mutator of  the  exciter,  and  part  is  taken  through  the  alter- 
nator field  rings  to  the  revolving  field  magnet,  and  part  goes 
through  a rheostat  to  the  stationary  exciter  fields.  So  far 
as  described  the  arrangement  corresponds  with  an  ordinary 


i 


114 


ALTERNATING  CURRENTS 


separately  excited  rotating  field  alternator  having  the  exciter 
mounted  upon  its  shaft.  In  order  to  procure  the  compounding 
effect,  however,  the  alternating  currents  from  the  secondary 
windings  of  three  series  transformers  placed  in  series  with  the 
three-phase  alternator  leads,  are  led  by  means  of  the  special 
exciter  armature  rings  into  the  exciter  armature  winding.  As 
explained  above,  the  exciter  and  alternator  field  magnets  have 
the  same  number  of  poles,  and  hence  the  frequency  of  the  electro- 
motive force  is  the  same  in  the  exciter  and  alternator  armatures, 
and  therefore  the  alternating  current  will  pulsate  through  the  ex- 
citer armature  with  the  peaks  of  its  waves  in  a definite  position 
with  reference  to  the  exciter  pole  pieces.  By  a proper  organi- 
zation, the  reactions  of  the  three-phase  currents  thus  fed  to  the 
exciter  armature  can  be  caused  to  increase  the  strength  of  the 
exciter  fields  in  a degree  dependent  on  both  the  amount  and 
phase  position  of  the  currents.  The  field  magnet  of  the  exciter 
is  mounted  so  that  it  can  be  angularly  shifted,  slightly,  so  as 
to  bring  the  impulses  of  the  alternating  current  into  proper 
relation  with  the  polar  positions.  Such  machines  can  be  made 
for  any  number  of  phases. 

34.  Rectifying  Commutators.  — The  rectifying  commutator 
has  as  many  segments  as  there  are  poles  on  the  alternator,  and 
alternate  segments  are  joined  in  electrical  connection,  making 

two  sets  composed  of  al- 
ternate segments,  as  illus- 
trated in  Fig.  101.  When 
series  excitation  is  to  be 
provided  by  the  rectified 
current,  one  terminal  of  the 
armature  winding  and  one 
terminal  of  the  external 
circuit  are  attached  respec- 
tively to  these  two  sets  of  segments,  as  shown  in  the  figure, 
or  the  secondary  terminals  of  a series  transformer  are  respec- 
tively attached  to  the  two  sets  if  the  excitation  is  to  be 
obtained  through  the  medium  of  such  a transformer,  as  shown 


Fig.  101.  — Rectifying  Commutator. 


in  Fig.  92.  Brushes  bearing  upon  the  commutator  at  non- 
sparking points  (that  is,  under  the  circumstances  considered, 
as  nearly  as  practicable  at  points  of  zero  current)  then  collect 
a rectified  current.  The  brushes  are  shown  in  Fig.  101  bear- 


ARMATURE  AND  FIELD  WINDINGS  FOR  ALTERNATORS  115 


mg  upon  adjacent  commutator  segments,  but  they  may  ob- 
viously be  placed  on  the  commutator  so  as  to  be  any  odd  num- 
ber of  segments  apart.  It  is  evident  that  direct  current  will 
flow  in  the  series  field  between  the  brushes  B,  B when  the 
circuit  connections  are  those  indicated  in  Fig.  101,  and  the  ma- 
chine is  in  operation.  The  commutator  possesses  as  many 
segments  as  there  are  poles  in  the  field  magnet,  and  if  the 
brushes  are  properly  placed,  each  brush  will  pass  from  one  set 
of  commutator  bars  to  the  other  at  each  alternation  of  the  cur- 
rent. That  is,  the  polarities  of  the  segments  reverse,  but  the 
brushes  simultaneously  slip  from  one  set  of  segments  to  the 
next,  and  therefore  remain  of  fixed  polarity.  The  current  in 
the  circuit  of  the  series  field  is  unidirectional,  although  it  is 
electrically  in  series  relation  with  the  armature  and  external 
circuit,  and  the  current  is  alternating  in  them. 

To  obtain  the  maximum  effect,  the  brushes  must  pass  across 
the  insulation  between  the  commutator  segments  at  about  the 
time  when  the  current  in  the  armature  reverses.  Minimum 
sparking  will  occur  if  the  brushes  are  shifted  to  the  point  where 
the  current  is  zero  in  the  circuit  at  the  instant  each  brush 
breaks  contact  with  one  segment  as  it  slips  on  to  the  next.  If 
the  brushes  are  shifted  in  the  forward  direction,  commutation 
will  take  place  on  a rising  current,  which  is  likely  to  cause 
quite  serious  sparking.  The  character  of  the  current  in  the 
series  field  winding  manifestly  could  be  modified  by  shifting 
the  position  of  the  brushes,  if  sparking  were  not  prohibitive. 
For  instance,  if  the  brushes  could  be  maintained  in  the  position 
where  commutation  takes  place  when  the  armature  current  is 
maximum,  the  current  in  the  series  field  circuit  would  in  each 
period  change  gradually  from  a maximum  in  one  direction  to  a 
maximum  in  the  other  and  be  then  reversed  to  a maximum  in 
the  first  direction,  and  the  resultant  effect  on  the  strength  of 
the  field  magnet  would  be  zero.  On  the  other  hand,  if  the 
brushes  are  upon  the  neutral  points,  that  is,  commutate  at  the 
instant  the  armature  current  is  zero,  the  current  is  unidirec- 
tional in  the  field  circuit. 

Various  devices  have  been  employed  to  avoid  sparking  at  the 
rectifying  commutator,  but  in  American  machines  no  special 
precautions  are  taken.  In  a Zipernowsky  alternator,  built 
by  Ganz  and  Co.  of  Budapest,  the  following  arrangement  of 


lie 


ALTERNATING  CURRENTS 


the  commutator  was  employed.  Between  the  commutator 
divisions  are  inserted  narrow  metallic  sectors  which  are  con- 
nected together.  Four  brushes  are  used,  two  on  each  side  of 
the  commutator.  One  brush  of  each  pair  is  set  a little  in  the 
lead  of  the  other,  and  the  pair  is  connected  together  through  a 
small  resistance.  The  leading  brushes  are  connected  directlv 
to  the  circuits.  When  the  commutating  point  is  reached  dur- 


Fig.  102.  — Special  Rectifier  for  Minimum 
Sparking. 

self-inductance  tends  to  uphold 
and  this,  therefore,  does  not  f; 
tion,  as  shown  in  Fig.  103,  hut 
wavy  line  more  like  that  of  Fig 


mg  the  rotation  of  the  com- 
mutator, the  trailing  brushes 
move  on  to  intermediate  seg- 
ments, while  the  forward 
brushes  are  still  on  main  seg- 
ments. Hence  both  the  field 
circuit  and  supply  circuit 
are  short-circuited  for  an  in- 
stant through  the  resistances 
connecting  the  brushes  (Fig. 
102).  Upon  short-circuit- 
ing the  fields  with  this  or  an 
ordinary  commutator,  their 
the  current  in  the  windings, 
ill  to  zero  at  each  commuta- 
the  current  curve  becomes  a 
. 101.  Picou  says  * that  it  is 


Fig.  103.  — Current  Impulses  that  would 
be  produced  by  a Rectifier  having  no 
Self-inductance  in  Circuit. 


Fig.  101.  — Curve  showing  the  Wavy  Cur- 
rent which  passes  from  a Rectifier  to 
the  Alternator  Fields. 


preferable  to  place  the  brushes  so  that  commutation  occurs 
slightly  earlier  than  the  point  of  least  sparking.  In  this  case 
the  spark  is  due  to  a decreasing  current,  and  is  thin  and  weak. 
With  the  commutation  occurring  later  than  the  point  of  least 
sparking,  the  spark  is  due  to  a rising  current,  and  it  is  of  great 


* Machines  Dynamo-lSlectriques,  p.  99. 


ARMATURE  AND  FIELD  WINDINGS  FOR  ALTERNATORS  117 


magnitude.  The  advantage  of  a wavy  current  in  the  field,  in 
stead  of  a discontinuous  one,  is  sufficiently  well  gained  by  the  use 
of  copper  brushes  of  considerable  thickness,  on  an  ordinary  recti- 
fying commutator,  which  short-circuit  the  supply  circuit  and  field 
circuit  at  the  instant  when  they  bridge  over  the  insulation  be- 
tween two  segments.  Figure  101  very  closely  represents  the 
form  of  current  curve  under  these  circumstances. 

Figure  101  shows  the  circuit  connections  at  the  rectifying 
commutator  of  a compound-wound  machine  with  rotating  arma- 
ture. For  a machine  with  a rotating  field  magnet,  the  series 
field  connections  and  the  connections  to  armature  and  external 
circuit  are  interchanged  so  that  the  former  go  directly  to  the 
commutator  segments  and  the  latter  to  the  brushes. 

35.  Inductor  Alternators.  — The  windings  of  inductor  alter- 
nator’s may  be  made  entirely  stationary,  thus  avoiding  collect- 
ing devices.  These  devices,  however,  are 
of  so  little  expense  in  construction  and 
material  that  there  is  no  marked  advan- 
tage in  suppressing  them,  and  the  mechani- 
cal and  magnetic  difficulties  encountered 
in  the  design  and  construction  of  inductor 
alternators  have  not  permitted  them  to 
come  into  general  use,  though  there  are 
several  theoretical  points  of  advantage 
presented  in  their  design.  In  order  to 
avoid  excessive  losses  due  to  eddy  cur- 
rents and  hysteresis  it  is  important  that 
the  magnetic  density  in  the  field  magnets 
be  kept  as  uniform  as  possible.  Since  this 
cannot  be  fully  accomplished,  it  is  neces- 
sary to  thoroughly  laminate  the  iron  in 
which  the  density  varies.  The  inductor 
must  be  moved  in  such  a way  as  to  periodi- 
cally short-circuit  or  break  the  lines  of 
force  which  naturally  pass  through  the  ar- 
mature coils.  This  may  be  accomplished 
as  shown  in  Fig.  105,  where  the  effective  reluctance  of  the 
total  magnetic  circuit  is  fairly  constant  for  all  positions  of  the 
inductor.  In  this  example  it  is  seen  that  as  an  inductor  travels 
from  one  pole  piece  to  the  next  of  opposite  sign,  the  mag- 


Fig.  105.  — Diagram  of 
an  Inductor  Alternator. 
A,  A are  the  Armature 
Windings,  NS  the  Field 
Windings,  and  BB  In- 
ductors. 


118 


ALTERNATING  CURRENTS 


netism  through  the  armature  coil  changes  from  a maximum  in 
one  direction  to  a maximum  in  the  other.  In  spite  of  the  fact 
that  the  field  poles  are  maintained  at  a fairly  constant  magnetic 
density  it  is  evidently  necessary,  in  order  to  sufficiently  reduce 
the  iron  losses,  to  laminate  all  the  iron  of  the  magnetic  circuit. 
The  figure  shows  that  the  reluctance  cannot  remain  entirely 
constant,  and  that  the  effective  ampere  turns  in  the  magnetic 
circuits  also  vary  with  the  position  of  the  inductor.  In  Figs. 
106  and  107  are  shown  two  types  of  inductor  machines  in 

which  no  attempt  is 
made  to  keep  the 
magnetic  circuit  of 


constant  reluctance. 
Each,  of  these  forms 
gains  some  economy 
in  construction  by 
uniting  the  coils. 

In  the  Stanley  al- 
ternators which  were 
at  one  time  consid- 
erably used  in  this 
country,  lines  of 
force  are  caused  by 
the  motion  of  the  inductor  to  sweep  across  the  armature  coils, 
while  the  total  magnetism  in  the  inductor  remains  fairly  con- 
stant.  The  field 
windings,  which  are 
arranged  in  the  form 
of  a single  coil,  em- 
brace the  inductor 
core,  and  though 
this  coil  is  stationary 
the  machines  are 
not  true  inductor 
alternators.  Figure 
108  shows  the  so- 
called  inductor  and 
field  coil  of  a 2000- 
kilowatt  alternator  of  the  Stanley  type.  The  field  coil  rests  in 
the  frame,  as  seen  in  the  figure,  and  does  not  rotate  with  the  in- 


f-ARMATURE  COIL 


Fig.  107. — Diagram  of  an  Inductor  Alternator  having  a 
Single  Magnetizing  Coil  and  Single  Armature  Coil. 


ARMATURE  AND  FIELD  WINDINGS  FOR  ALTERNATORS  119 


Fig.  108.  — Inductor  and  Field  Coil  of  a 2000-kilowatt  Stanley  Alternator. 


Fig.  109.  — Illustration  show- 
ing the  Armature  Windings 
of  a 2000-kilowatt  Inductor 
Alternator.  Two  Coils  are 
partly  Removed. 


ductor.  This  coil  magnetizes  the  inductor  so  that  the  crown  of 
inductor  projections  on  one  side  of  the  coil  are  of  one  sign  and 
those  on  the  other  are  of  the  opposite 
sign.  The  armature  coils  are  usually  of 
the  distributed  type,  as  shown  in  Fig. 

109,  and  in  modern  machines  are  always 
chord  wound.  The  armature  windings 
are  also  in  two  crowns ; one  surround- 
ing each  crown  of  inductor  projections. 

There  are  in  each  phase  twice  as 
many  coils  in  each  crown  as  inductor 
projections,  and  in  order  that  they 
shall  add  their  voltages  the  coils  must 
be  connected  into  circuit  alternately  right  and  left  handed. 

The  Warren  alternator  was  very  similar  to  the  Stanley  ma- 
chine. It  had,  however,  only  one 
set  of  inductors  and  armature  coils. 
Figure  110  shows  the  inductor. 
The  armature  windings  were  placed 
on  the  inner  surface  of  a laminated 
stationary  iron  frame  surrounding 
the  projections  at  the  left  hand 

gle  Crown  of  Projections.  °f  the  fiSU1'e  ’ and  the  magnetiz- 
ing coil  surrounded  the  cylindrical 
portion  at  the  right.  The  magnetic  circuit  was  completed 
through  the  frame. 


120 


ALTERNATING  CURRENTS 


36.  Armature  Insulating  and  Core  Materials  and  Construction. 

— Before  the  conductors  are  placed  on  an  armature  core  it  is 
usual  to  insulate  it,  very  much  as  in  the  case  of  a direct-current 
armature,*  but  more  thoroughly  on  account  of  the  high  voltages 
usually  produced  in  alternator  armatures.  For  this  purpose 
mica,  micanite,  mica  paper  and  cloth,  shellacked  canvas,  fuller 
board,  oiled  paper  and  cloth,  sheets  of  vulcanite,  vulcabeston,  vul- 
canized fiber,  and  similar  insulating  materials,  are  used.  Mica, 
micanite,  vulcanite,  vulcabeston,  bonsilate,  vulcanized  fiber, 
asbestos  paper,  and  similar  materials  are  also  used  to  insulate 
collecting  rings  and  brush  holders,  and  for  insulation  between 
the  armature  coils.  The  wire  used  for  high-voltage  alternator 
armatures  is  often  triple-cotton  covered,  and  is  thoroughly 
japanned  during  the  process  of  winding.  Vulcanized  fiber  is 
made  from  paper  fiber  by  a chemical  process  and  is  furnished 
in  sheets  and  tubes.  Its  convenient  form,  cheapness,  and  ease 
of  working  have  brought  it  into  extensive  use.  It  unfortu- 
nately absorbs  moisture  when  exposed  to  the  air,  which  causes 
it  to  expand  and  contract  to  a remarkable  degree.  The  mois- 
ture also  reduces  its  insulating  qualities  to  a large  extent.  It 
is  therefore  unsafe  to  place  entire  reliance  upon  it  where  con- 
tinuously high  insulation  is  required.  Of  the  various  available 
insulating  materials  mica  is  the  only  thoroughly  reliable  one, 
but  it  is  unduly  expensive  and  of  poor  mechanical  qualities. 
On  the  latter  account  it  is  generally  used  in  combination  with 
other  materials.  When  made  up  in  the  form  of  micanite  by 
treating  with  varnish,  laying  in  overlapping  scales  between 
paper  or  cambric,  and  subjecting  to  high  pressure  and  heat,  its 
mechanical  qualities  are  somewhat  improved,  and  it  may  be 
formed  into  sheets  and  tubes  or  molded  as  desired.  Materials 
such  as  vulcabeston  and  bonsilate  are  advantageous  for  insulat- 
ing collector  rings  and  similar  details,  since  they  can  be  molded 
into  any  desired  form.  Vulcabeston,  which  is  a compound  con- 
taining rubber  and  asbestos,  is  also  manufactured  in  sheets  and 
has  quite  satisfactory  mechanical  and  electrical  properties. 
Bonsilate  is  not  sufficiently  tough  for  very  general  service. 
Boxwood  and  paraffined  maple  or  hickory  are  frequently 
used  for  insulation  where  considerable  bulk  is  required  and 
their  mechanical  properties  will  serve,  though,  as  they  are  liable 
* Jackson’s  Electromagnetism  and  the  Construction  of  Dynamos,  p.  104. 


ARMATURE  AND  FIELD  WINDINGS  FOR  ALTERNATORS  121 


to  be  more  or  less  affected  by  the  surrounding  condition  of 
humidity,  they  must  be  used  with  care.  A special  material 
called  asbestos  wood  is  now  manufactured  in  sheets  and  plates 
adapted  for  use  as  insulating  material  in  some  of  the  situations 
now  unsatisfactorily  filled  by  other  materials. 

Tubes  or  troughs  of  micanite,  varnished  fiber,  or  some  other 
insulator  that  will  meet  the  conditions,  are  ordinarily  slipped 
into  the  armature  slots  before  the  wires  are  inserted.  The  coils, 
after  careful  taping,  are  then  placed  within  the  tubes.  The 
method  of  insulating  armature  conductors  is  illustrated  in  Figs. 
79  to  83.  When  a coil  is  made  up  of  many  turns  of  wire,  it  is 
sometimes  necessary  to  also  insulate  between  layers,  in  addition 
to  the  insulation  afforded  by  the  cotton  covering  of  the  wire 
itself.  The  field  coils  of  alternators  are  insulated  with  the 
same  materials  as  the  armature  windings,  though  the  lower 
voltages  usually  employed  in  the  field  circuits  call  for  a less 
thickness  of  insulation. 

The  field  coils  are  generally  wound  upon  formers,  and  insu- 
lating sheets  are  placed  between  layers.  The  wire  used  is  often 
double  cotton  covered ; but  a flat  strip  (sometimes  bare)  is  not 
uncommonly  used.  This  latter  may  be  wound  on  edge  with 
insulating  strips  between  the  conducting  strips.  Winding  a 
single  layer  of  strips  on  edge  is  thought  to  improve  the 
mechanical  and  electrical  stability  of  the  field  coil  and  improve 
its  heat-dissipating  qualities.  The  taped  coil  may  be  placed 
upon  an  insulating  spool  and  then  slipped  upon  the  pole  piece. 
Formed  conductors  are  often  given  a coating  of  enamel  for 
insulation;  and  coils  are  frequently  heated  to  very  high  tempera- 
ture during  construction,  while  under  the  high  pressure  of  a 
former,  to  drive  off  volatile  matter  and  leave  the  conductors 
embedded  in  a compact  and  stable  mass  of  insulation. 

In  insulating  alternators  care  must  be  taken  to  use  such 
materials  as  will  not  deteriorate  either  electrically  or  mechani- 
cally under  the  rather  high  temperature  which  the  machines 
attain  when  running.  If  the  insulating  material  softens  per- 
ceptibly, it  may  permit  points  of  high  difference  of  potential  to 
come  close  together  with  a resultant  short  circuit ; or  the  same 
failure  may  result  in  the  permanent  lowering  of  the  specific 
resistance  of  the  insulating  materials. 

The  Dielectric  strength,  or  the  property  of  an  insulator  to  resist 


122 


ALTERNATING  CURRENTS 


electric  puncture,  is  the  basis  upon  which  to  calculate  insula 
tion,  rather  than  the  electrical  resistance,  though  the  latter  often 
serves  as  an  indication  of  the  former.  The  following  table 
gives  the  specific  resistance,  or  the  resistance  in  megohms  per 
centimeter  of  length  and  square  centimeter  of  cross  section,  of 
familiar  insulating  materials  at  ordinary  temperatures  (namely, 
about  70°  F.).  The  values  are  only  approximate,  as  different 
samples  of  the  same  material  are  apt  to  have  quite  different 
characteristics. 

TABLE 


Specific  Resistance  of  Insulators  (in  megohms) 


Mica 

Paper 

Paraffine  oil 

Rubber 

Shellac 

Yulcanized  fiber 


10«  to  108 
104  to  105 
106  to  108 

108  to  109 

109  to  1010 
106  to  108 


The  dielectric  strength  of  any  insulating  material  is  quite  de- 
pendent upon  the  character  of  the  particular  sample  under  test, 
the  character  of  the  electrodes,  the  duration  of  the  stress,  and 
other  conditions  surrounding  the  tests;  and  therefore  the  data 
from  tests  should  only  be  used  for  rough  estimates  or  checks. 
Dr.  Steinmetz  * has  worked  out  a number  of  special  formulas 
from  the  results  of  tests  which  may  be  used  in  this  way.  In  the 
following  table  of  these  formulas,  d is  the  thickness  of  dielec- 
tric in  milli-centimeters  through  which  a disruptive  discharge 
will  occur  under  the  impulsion  of  E kilovolts.  The  given 
voltage  is  the  maximum  value  of  a sinusoidal  voltage  wave 
between  flat  electrodes. 


dielectric  materials 
Air  .... 
Mica  .... 
Yulcanized  Fiber 
Dry  Wood  Fiber 
Paraffined  Paper  . 

Melted  Paraffine  . 

Copal  Varnish 
Crude  Lubricating  Oil 
Vulcabeston 
Asbestos  Paper 


TABLE 

FORMULAS 

. d = 36  (e~1SE  — 1)  + 54  E + 1.2  E2 
. d = .24  E + .0115  E2 
. d = 7.66  E + 2.3  E2 
. d = 7.66  E 
. d = 3 E 
. d = 12.4  E 
. d = 30  E 
. d = 60  E 
. d = 28E 
. d = 23  E 


* Steinmetz  on  Dielectrics,  Trans.  Amer.  Inst.  Elect.  Eng.,  vol.  10,  p.  85. 


ARMATURE  AND  FIELD  WINDINGS  FOR  ALTERNATORS  123 


In  obtaining  data  upon  which  these  very  rough  formulas  are 
based,  alternating  currents  of  about  100  cycles  per  second  were 
used,  and  the  voltage  was  varied  from  two  to  thirty  thousand 
volts.  In  general,  thin  layers  of  dry  air  withstand  about  six 
thousand  volts  per  millimeter  when  the  spark  occurs  between 
needle  points;  oils  from  twelve  hundred  to  a hundred  thousand 
volts,  and  specially  selected  oils  from  one  quarter  to  a million 
volts  per  centimeter  between  needle  points,  depending  upon  the 
quality ; and  mica  from  several  hundred  thousand  to  several 
million  volts  per  centimeter  between  flat  electrodes  pressing 
the  material  between  them. 

The  wire  used  for  winding  the  fields  and  armatures  of  alter- 
nators is  almost  always  double  or  triple  cotton-covered  copper 
wire  or  strip  of  high  conductivit}'.  The  standard  of  conduc- 
tivity adopted  by  the  American  Institute  of  Electrical  Engi- 
neers in  1893  is  9.59  ohms  per  mil-foot  at  0°  C.*  The 
average  rate  of  change  of  resistance  per  centigrade  degree  of 
change  of  temperature  within  ordinary  ranges  may  be  taken 
as  .42  per  cent  of  the  resistance;  at  0°  C.  This  makes  the 
resistance  at  any  temperature,  in  comparison  with  the  resist- 
ance at  0°  C., 

Rt  = R0(l  + .0042  0, 

where  Rt  is  the  resistance  at  the  required  centigrade  tempera- 
ture, R0  at  zero  temperature,  and  t is  the  required  temperature. 
The  copper  conductors  used  in  alternator  windings  should  fall 
little  below  this  standard.  For  winding  armatures  and  fields 
the  copper  is  made  up  in  the  form  of  bars  (Fig.  83),  strap 
(Figs.  80  and  81),  or  the  ordinary  round  wire  (Fig.  79),  as  the 
conditions  make  desirable.  In  large  machines  bar  or  strap 
copper  is  used  almost  exclusively. 

The  armature  cores  themselves  may  be  made  of  punched  iron 
disks,  iron  punchings  of  special  shapes,  iron  wire,  or  iron  tape. 
In  American  machines  the  punchings  are  almost  universal. 
The  old  Ivapp  machines  had  armature  cores  made  of  iron'  tape.f 
In  the  larger  sizes  of  American  machines,  which  are  built  to 
connect  directly  to  steam  engines  or  large  turbines,  and  there- 

* Trnns.  Amer.  Inst.  Elect.  Eng.,  October,  1893.  See  also  Standardization 
Rules  of  1907,  Par.  260. 

t Kapp’s  Dynamos,  Alternators,  and  Transformers,  p.  467. 


124 


ALTERNATING  CURRENTS 


fore  have  armatures  of  large  diameters,  the  armature  cores 
are  usually  built  up  of  segmental  punchings  put  together  in 
such  a way  that  the  segments  of  alternate  layers  break  joints 


Fig.  111.  — Punching  for  Rotating  Alternator  Armature. 

(Fig.  111).  In  stationary  armatures  the  teeth  are,  of  course, 
inside  and  the  keys  outside. 

Figure  112  shows  an  old-style  single-phase  machine  with 
rotating  armature  constructed  with  one  armature  slot  per  field 
pole.  At  A'  may  be  seen  one  of  the  punchings  which  go  to 
make  up  the  laminations  of  the  armature.  The  field  pole 
pieces  are  solid  in  this  machine,  which  is  common,  though  they 


Fig.  Ill  a.  — Core  Stamping  with  Distance  Pieces  for  Procuring  Ventilation. 

are  also  sometimes  made  up  of  punchings  secured  in  the  frame 
in  numerous  ways. 

It  is  usual  to  provide  ventilating  ducts  between  core  lamina- 
tions, and  these  ducts  are  provided  for  in  the  assembling  by  the 
insertion  of  distance  pieces.  The  distance  jueces  c , c , illustrated 
in  Fig.  Ill  a are  usually  ribs  of  brass  or  T angle  iron  which  may 
be  one  half  inch  or  more  in  depth  in  large  size  machines  and 
are  riveted  to  core  stampings.  These  are  placed  at  the  desired 
distance  apart  in  the  core  and  make  annular  ventilating  ducts 
at  intervals  of  a few  inches  in  the  length  of  the  core.  The 


ARMATURE  AND  FIELD  WINDINGS  FOR  ALTERNATORS  125 


ventilating  ducts  are  to  be  plainly  seen  where  they  come  to  the 
face  of  the  armature  core  in  various  illustrations  of  the  text, 
such  as  Figs.  80  to  84. 

The  sectional  view  in  Fig. 

113  also  shows  the  ventilat- 
ing ducts. 

Figure  113  is  a partial 
side  elevation  and  section 
of  a rotating-field  alterna- 
tor having  a three-phase 
armature  winding  with 
one  slot  per  pole  per  phase, 
in  which  the  general 
method  of  mechanical  con- 
struction is  shown.  Much 
care  must  be  taken  in  the 
selection  of  iron  for  the 
magnetic  circuit  of  alter- 
nators, in  order  to  keep 
the  iron  losses  at  a mini- 
mum. This  subject  will  be  discussed  at  some  length  in 
a later  chapter.* 


Fig.  112. — Side  Elevation  of  Alternator, 
showing  Armature  Punchings. 


Fig.  113.  — Side  Elevation  and  Section  of  Rotating  Field  Alternator,  showing 
General  Construction. 


In  high-speed  turbine  generators  the  rotating  field  magnet 
must  be  made  very  substantial  to  withstand  the  strains  due  to 


* Chap. IX. 


126 


ALTERNATING  CURRENTS 


Fig.  113  a.  — Two-pole  Ro 
tating  Field  for  a Tur 
bine  Generator. 


the  high  circumferential  velocity.  Figure  113  a shows  a West- 
inghouse  two-pole  rotating  field  magnet  with  the  lower  half  of 
the  field  partly  wound.  After  the  coils 
have  been  wound  in  the  slots  and  insu- 
lated, as  shown  in  the  lower  half  of  the 
figure,  they  are  fastened  firmly  in  place 
by  heavy  metal  strips  driven  through  the 
keyways  shown  near  the  outer  ends  of 
the  lugs.  Thus  the  field  is  made  entirely 
metal-clad.  Ventilation  is  obtained  by 
means  of  the  holes  seen  in  the  ends  of 
the  core  and  by  the  circumferential  slots.  The  core  itself  is 
built  up  of  steel  disks  about  .015  of  an  inch  thick,  which  are 
machined  for  the  windings  and  ventilation. 

Figure  113  b shows  an  alternator  of  the  General  Electric  Co. 
on  a vertical  shaft,  driven  by  a Curtis  steam  turbine.  This 
figure  shows  the  machine  in  one  half  section.  The  nozzles  and 
guide  vanes  of  the  turbine  are  within  the  lower  cylindrical  por- 
tion. The  shaft,  which  carries  the  multipolar  rotating  field 
magnet  of  the  alternator  at  its  upper  part  and  the  turbine 
runners  at  its  lower  part,  is  supported  on  a step-bearing  sup- 
plied with  water  under  high  pressure  as  a lubricant. 

The  fields  and  armature  of  the  Westinghouse  alternator  illus- 


trated in  Figs.  83  a and  113  a are  intended  to  be  driven  by  a 
horizontal  steam  turbine  of  the  Westinghouse-Parsons  type 
which  runs  at  a high  speed,  and  the  field  has  fewer  poles,  — in 
this  case  it  is  bipolar. 


* See  Arts.  106  and  113. 


ARMATURE  AND  FIELD  WINDINGS  FOR  ALTERNATORS  12 


Fig.  113  6.  — Alternator  on  Vertical  Shalt  of  a Steam  Turbine. 


CHAPTER  IV 


SELF-INDUCTION,  CAPACITY.  REACTANCE.  AND  IMPEDANCE 


37.  Self-induction.  — Before  proceeding  with  the  develop- 
ment of  the  design  and  operation  of  alternating-current  ma- 
chines, it  is  essential  to  discuss  the  relations  which  exist 
between  voltages  and  currents  in  circuits  which  carry  alternat- 
ing currents.  It  is  to  be  remembered  that  the  current  pro- 
duced by  cutting  lines  of  force  sets  up  in  turn  magnetic  force, 
which  is  in  opposition  to  the  original  field.  Again,  when  a 
current  is  introduced  into  a circuit,  it  produces  a magnetic 
flux  the  rise  of  which  causes  a counter-voltage.  These  condi- 
tions are  necessary  under  the  law  of  conservation  of  energy 
and  its  corollary,  Lenz’s  Law.*  This  counter-voltage  is  called 
the  Voltage  or  Electro-motive  force  of  Self-induction,  and  the 
phenomenon  as  a whole  is  called  Self-induction. 

Self-induction , therefore , may  he  defined  as  the  inherent  quality 
of  an  electric  circuit  whereby  the  electro-magnetic  induction  set  up 
by  an  electric  current  in  the  circuit  tends  to  impede  the  current's 
own  introduction , variation , or  extinction  in  the  circuit. 

The  voltage  in  a circuit  which  at  any  instant  is  due  to  self- 
induction  is  evidently  proportional  to  the  rate  of  change  of  the 
magnetic  flux  set  up  by  the  current  of  the  circuit ; therefore, 
the  voltage  of  self-induction  is  proportional  to  the  rate  of 
change  of  current  in  the  circuit,  provided  that  the  reluctance 
of  the  magnetic  circuit  remains  constant  ; or 


ez  ce- 


de}) _di 
dtX~Jf 


where  et  is  the  instantaneous  self-induced  voltage  in  the  circuit, 
ej)  the  magnetism,  and  i the  current,  respectively,  at  the  same 
instant  of  time  t..  The  minus  sign  is  used  because  the  voltage 
induced  at  each  instant  is  in  a direction  to  impede  the  changes 
of  current  and  magnetism. 

* Jackson’s  Electro-magnetism  and  the  Construction  of  Dynamos,  p.  77. 

128 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  129 


If  the  circuit  exhibits  the  phenomena  of  electrical  resistance 
and  self-induction,  and  none  other,  as  would  be  true  of  a sim- 
ple coil  of  wire,  the  voltage  impressed  on  its  terminals  must  at 
each  instant  be  sufficient  to  overcome  the  counter- voltage  e:  and 
supply  the  iR  drop  corresponding  to  the  current  at  the  par- 
ticular instant.  That  is,  the  voltage  impressed  on  the  termi- 
nals of  the  circuit,  or  Impressed  voltage,  is  made  up  of  two 
components,  one  equal  and  opposite  to  the  counter-voltage  and 
the  other  equal  to  the  drop  of  voltage  through  the  resistance 
of  the  circuit.  The  latter  component  represents  the  portion  of 
the  force  driving  the  current  through  the  circuit  which  results 
in  expending  power  therein. 

When  the  instantaneous  values  of  current  and  voltage  are 
considered,  the  iR  drop,  the  counter-voltage  eh  and  the  im- 
pressed voltage  e are  scalar  values,  and  therefore  iR  — e — e, ; 
but  a little  consideration  will  show  that  the  Phase  of  the  alter- 
nating' voltage  of  self-induction  is  not  in  unison  with  that  of 
the  current,  although  their  periods  are  the  same  ; because  the 
counter- voltage  is  proportional  to  the  rate  of  change  of  magnet- 
ism caused  by  the  current,  and,  supposing  the  magnetism  to  be 
alternating,  its  rate  of  change  is  zero  when  it  is  at  a maximum, 
and  the  voltage  of  self-induction  is  therefore  zero  at  the  same 
time.  Manifestly,  therefore,  a vector  relation  must  be  recog- 
nized when  the  effective  values  of  current  and  voltage  are 
considered. 

When  the  curve  of  current  is  a sinusoid,  the  rate  of  change 
of  its  ordinates  at  any  point  is 


im  being  the  maximum  value  of  current ; and  the  counter-electro- 


assuming  that  the  reluctance  of  the  magnetic  circuit  is  uniform. 
The  curve  sin  (a  + 90°)  is  manifestly  a curve  which  occupies  a 
position  90°  earlier  than  (or  in  the  Lead  of)  the  curve  sin  a,  and 
the  curve  — sin  (a  + 90°)  is  180°  later  than  sin  (a  + 90°)  and 
therefore  90°  later  than  the  curve  sin  a.  Hence  the  curve  of 
the  counter-voltage  of  self-induction  is  a sinusoid,  the  phase  of 
which  Lags  90°  behind  (that  is,  is  90°  later  than)  the  phase 


di  _ imd (sin  a) 
dt  dt 


motive  force  et  is  therefore  proportional  to  — 


sin  ( a -f  90 °)da 
dt 


130 


ALTERNATING  CUR R ENTS 


of  the  current  setting  it  up.  It  is  evident  also  that  this  must  be 
the  case,  since  when  sinusoidal  current  is  passing  through  the 
zero  value,  rising  in  the  positive  direction  of  flow  through  the 
circuit,  its  rate  of  change  is  a maximum  and  the  self-induced 
voltage  is  at  maximum  value  in  a direction  opposing  the  rise 
of  current ; and  when  the  current  has  risen  to  a maximum  (a 
quarter  period,  or  90°,  later),  its  rate  of  change  is  zero,  and  the 
self-induced  voltage  is  zero  and  about  to  change  in  direction 
from  negative  to  positive.  Thus  it  is  seen  that  the  self-induced 
voltage  goes  through  its  cycle  in  a phase  which  is  90°  later 
than  the  current. 

38.  Vector  Diagrams  representing  the  Voltage  Relations  in  an 
Inductive  Alternating-current  Circuit.  — Suppose  the  line  00 


C describes  a circle  and  its  projection  on  the  vertical  axis 
describes  a simple  harmonic  motion.  (In  this  book,  such  vec- 
tors are  assumed  to  rotate  in  a counter-clockwise  direction.) 
At  each  instant  the  projection  of  the  line  on  the  vertical  axis 
proportionally  represents  the  magnitude  of  the  instantaneous 
active  voltage  (i.e.  iR  drop)  corresponding  to  the  angular 
advance  of  the  line.  The  projection  of  the  line  OjD,  which  is 
90°  behind  00,  likewise  represents  the  instantaneous  counter- 
voltage  at  each  instant.  The  algebraic  sums  of  the  instanta- 
neous  projections  of  OC  and  CA  (the  reverse  of  OR')  are  al 
ways  equal  to  the  simultaneous  projections  of  OA,  where  OA 
is  the  impressed  voltage.  The  impressed  voltage  OA  has  the 
magnitude  and  phase  relation  of  the  resultant  of  the  active 
voltage  (iR  drop)  and  the  opposite  of  the  self-induced  voltage. 
This  may  be  expressed  in  still  another  way,  which  may  pos- 
sibly give  a more  useful  conception  : The  impressed  voltage  ap- 

plied to  an  alternating-current  circuit  must  have  such  magnitude 


Fig.  114.  — Phase  Diagram  of  Voltages  in  a Self-inductive 
Circuit. 


(Fig.  114)  repre- 
sents the  maxi- 
mum value  of  the 
iR  drop  in  a cir- 
cuit carrying  a 
sinusoidal  cur- 
rent. If  the  line 
is  uniformly  ro- 
tated around  the 
point  0,  its  end 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  131 


and  jjhase  relation  that  it  will  neutralize  all  other  live  voltages 
and  at  the  same  time  furnish  the  active  voltage  which  drives  the 
current  through  the  resistance  of  the  circuit.  From  the  relations 
given  it  is  seen  that 

( OAf  = (OCf  + ( GA)\  or  OA  = V(<967)2+  ( CA f. 

The  lengths  of  the  lines  in  the  figure  have  been  assumed 
to  be  proportional  to  maximum  values  of  the  voltages,  but  the 
effective  values  of  the  voltages  hold  the  same  relations  since 
each  effective  value  is  equal  to  the  maximum  multiplied  by 
.707.  Consequently,  the  effective  value  of  the  impressed  vol- 
tage operating  in  a circuit  with  resistance  and  self-inductance  in 
series  is  equal  to  the  square  root  of  the  sum  of  the  squares  of 
effective  values  of  the  active  voltage  and  the  inductive  voltage 
reversed.  Or  when  E,  Er,  and  Ex  are  respectively  effective 
values  of  impressed  voltage,  active  voltage,  and  self-induced 
voltage  reversed, 

E=fE2  + Ef. 

The  triangle  of  voltages  representing  these  conditions  is  shown 
in  Fig.  115.  (Such  triangles,  in  this  book,  usually  have  the 
active  voltage  line  laid  out  in  a 
horizontal  direction.) 

If  the  heights  of  the  points  A, 

(7,  and  D of  the  uniformly  rotat- 
ing vectors  OA,  OC , and  OD  are 
plotted  as  ordinates  to  the  corre- 

, . . - . Fig.  115.  — Vector  Polygon  of  Vol- 

spondmg  talues  ol  time  01  of  tages  for  Series  Self-inductive  Cir- 
ce laid  out  on  the  axis  of  abscissas  cuit. 

as  illustrated  in  Figs.  3 and  4,  three  sine  curves  are  traced 
like  those  at  the  right  hand  of  Ffig.  114.  These  are  marked 
respectively  E , En  and  — Ex.  They  represent  the  waves  of 
impressed  voltage,  IR  drop,  and  inductive  voltage.  The  in- 
stantaneous value  of  the  impressed  voltage  at  each  instant  is 
equal  to  the  corresponding  instantaneous  values  of  Er  and  Ex 
added  together  algebraically. 

The  instant  of  time  in  the  sine  wave  plot  which  corresponds 
to  the  time  when  the  rotating  vectors  arrive  at  the  position  in- 
dicated by  the  left-hand  part  of  Fig.  114  is  shown  by  the 
dotted  vertical  axis  cutting  through  the  sine  curves.  If  the 
zero  of  time  is  assumed  at  the  instant  when  the  impressed  voltage 


132 


ALTERNATING  CURRENTS 


is  passing  through  zero  from  the  negative  to  the  positive  direc- 
tion, this  would  be  indicated  by  the  phase  diagram  of  rotating 
vectors  at  the  instant  when  OA  lies  in  the  positive  direction 
along  the  W-axis.  The  relations  of  the  voltage  waves  at  that 
instant  are  indicated  at  the  left-hand  terminus  of  the  plot  of 
sine  curves.  It  is  obvious  that  the  inductive  voltage  must  be 
equal  to  the  iR  drop  at  that  instant,  to  maintain  the  current  i, 
since  the  then  instantaneous  value  of  the  impressed  voltage  is 
zero. 

Prob.  1.  Construct  a vector  polygon  of  voltages  for  a circuit 
containing  a self-inductive  sinusoidal  voltage  of  50  volts  and 
an  IR  drop  of  50  volts. 

Prob.  2.  Construct  a vector  polygon  of  voltages  for  a circuit 
containing  an  IR  drop  of  100  volts  and  a self-inductive  sinu- 
soidal voltage  of  5 volts. 

Prob.  3.  Construct  a vector  polygon  of  voltages  for  a circuit 
containing  20  volts  IR  drop  and  500  sinusoidal  self-inductive 
volts. 

Prob.  4.  What  are  the  values  of  the  impressed  voltages  in 
problems  1,  2,  and  3 ? 

Prob.  5.  Draw  the  curves  of  voltage  in  a circuit  having  an 
IR  drop  (effective  value)  of  100  volts  and  a self-inductive 
voltage  (effective  value)  of  50  volts,  on  the  supposition  that 
the  curves  are  sinusoids. 

Prob.  6.  In  problem  5 find  the  instantaneous  values  of  the 
self-inductive  and  impressed  voltages  when  the  IR  drop  is  0 ; 
find  the  impressed  voltage  and  IR  drop  when  the  self-inductive 
voltage  is  0 ; find  the  IR  drop  and  self-inductive  voltage 
when  the  impressed  voltage  is  0. 

Prob.  7.  Find  the  instantaneous  values  of  the  three  voltages 
in  problem  5 when  a — 0,  45°,  90°,  135°,  180°,  225°,  270°,  315°, 
and  360°  ; find  these  values  graphically  from  the  rotated  vector 
polygon  and  from  the  curves  of  voltage. 

39.  Angle  of  Lag.  — The  angle  6 between  the  lines  OA  and 
OC  in  Figs.  114  and  115  shows  the  amount  by  which  the  phase 
of  the  active  voltage  (which  is  in  phase  with  the  current)  lags 
behind  that  of  the  impressed  voltage.  The  figure  makes  it  evi- 


LF-INDUCTION,  CAPACITY,  PEACTANCE,  AND  IMPEDANCE  133 

dent  that  this  lag  is  caused  by  the  position  of  the  self-inductive 
voltage,  and  that  its  magnitude  depends  upon  the  relative 
lengths  of  the  line  6hi(  = — OB')  and  the  line  00.  .The  tangent 
of  the  angle  is 

„ CA  Inductive  voltage  reversed 

tan  0 = — — — 

00  Active  voltage 

The  angle  0 is  called  the  Angle  of  lag.  The  relations  are 
plainly  shown  in  Fig.  115. 

Prob.  1.  What  are  the  angles  of  lag  in  problem  1,  problem  2, 
and  problem  3 of  Article  38  ? 

Prob.  2.  Find  the  angle  of  lag  in  a circuit  which  has  an 
impressed  sinusoidal  voltage  of  100  volts  and  an  active  sinu- 
soidal voltage  (IR  drop)  of  30  volts. 

Prob.  3.  Find  the  angle  of  lag  in  a circuit  which  has  an 
impressed  sinusoidal  voltage  of  300  volts  and  a self-inductive 
sinusoidal  voltage  of  150  volts. 


40.  Sinusoidal  Voltage  in  a Self-inductive  Circuit  expressed 
as  a Complex  Quantity.  — The  complex  quantity  which  ex- 
presses the  vector  representing  the  impressed  voltage  is 

E — Er  + 3 Ex, 

or  E = Z'fTos  6 +j  sin  d), 

where  E is  the  numerical  or  scalar  value  of  E , Er  the  scalar  value 
of  IR  drop  and  Ex  the  scalar  value  of  inductive  voltage  reversed. 


Also 


E = VE*  + E}  and  tan  6 = ^. 

E,. 


If  there  are  a number  of  circuits  in  series  having  impressed 
voltages  Ev  E2,  ZJ3,  etc.,  the  combined  voltage  across  the  out- 
side terminals  is  represented  by  the  vector  sum 


E — Er  +jEx  — (Eri  + Er2+Era  + etc. ~)  + j (EXi + EXi  + EXi  + etc.) 
in  which 


Er  — Eri  -f-  Ert  -f-  Eri  -(-  etc.  and  Ex  — EXi  + EXt  + EXa  -f-  etc. 
The  angle  of  lag  is  equal  to 

g=  tan-1  ^•ri  ^ + ^ etc-) 


(Zy1  + EVi  -(-  Zy3  -f-  etc.)  Er 


The  angle  6 is  the  angle  of  lag  measured  between  the  phase  of 


134 


ALTERNATING  CURRENTS 


the  current  flowing  in  the  circuit  and  the  voltage  impressed 
upon  the  entire  circuit. 

Prob.  1.  The  IR  drop  in  a circuit  is  50  volts  and  the. self- 
inductive  voltage  25  volts.  Find  the  impressed  voltage  and 
the  angle  of  lag  of  the  current.  (In  this  and  the  following 
problems  all  voltages  are  assumed  to  be  sinusoidal.) 

Prob.  2.  The  IR  drop  in  a circuit  is  50  volts  and  the  self- 
inductive  voltage  500  volts.  Find  the  value  of  the  impressed 
voltage  and  the  current  lag. 

Prob.  3.  The  angle  of  lag  in  a circuit  is  30°  and  the  im- 
pressed voltage  200  volts.  What  are  the  values  of  IR  drop 
and  self-inductive  voltage  ? 

Prob.  4.  The  IR  drop  and  self-inductive  voltage  of  two  cir- 
cuits are  respectively  50  and  30  volts,  and  40  and  20  volts. 
What  will  be  the  resultant  voltage  if  the  two  circuits  are  placed 
in  series,  and  what  will  be  the  angle  of  lag  between  the  current- 
flow  and  the  voltage  impressed  between  the  main  terminals  ? 

Prob.  5.  Two  circuits  in  series  have  voltages  impressed 
upon  them  of  100  and  200  volts  and  angles  of  lag  of  30°  and 
45°  respectively.  Calculate  the  total  voltage  across  the  two 
circuits  and  the  angle  of  lag  of  the  current  with  respect  thereto. 

Prob.  6.  Three  voltages  in  a circuit  of  100,  200,  and  300 
volts  are  in  series,  the  angles  of  lag  in  the  three  parts  of  the 
circuit  being  respectively  20°,  40°,  and  60°.  What  are  the 
resultant  voltage  and  angle  of  lag  ? Also  what  are  the  IR 
and  self-inductive  components  of  the  total  voltage  ? 

41.  Self-inductance.  — The  magnitude  of  the  voltage  of  self- 
induction  is  dependent  upon  the  number  of  lines  of  force 
inclosed  by  the  circuit  per  unit  current  flowing  in  it,  the  num- 
ber of  turns  composing  the  circuit,  and  the  current  flowing 
therein;  which  follows  from  the  fundamental  equation  for  vol- 
tage* ; the  formula  is 

ndd> 

e‘  ~ ~ l¥Jt' 

where  et  is  the  voltage  of  self-induction  for  the  given  rate  of 
change  of  magnetization,  n is  the  number  of  turns  in  the  coil. 


* Art.  ll. 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  135 


and,  as  before,  — is  the  rate  of  change  of  magnetism.  But  in 
dt 

a long  or  closed  solenoid,  the  magnetizing  force  is  * 

7i/T  _ 4 tmi 
10  ’ 


where  i is  the  value  of  the  current  at  the  instant  under  con- 
sideration; and  the  reluctance  of  the  magnetic  circuit  isf 


where  l and  A are  respectively  the  average  length  and  cross 
section  of  the  solenoid  and  y.  is  the  permeability,  assumed  in 
this  paragraph  to  be  uniform.  Therefore,  the  number  of  lines 
set  up  by  a current  i is  f 


and 


4 nrniAy 
10 1 ’ 


nd<f>  _ n ^ 4 irnAy 
108  dt  ~ ~ 108  X 10  z " 


j,  4:7rnAu  j. 

d+  = -wT  x*> 

, di  _ 4 irn2Ay  w di 

( jt~  io^  x it 


The  numerical  value  of 


n ^ 4 irnA/i 

To8  x 10  l 


is  called  the  Self-inductance  or  the  Coefficient  of  self-induction 

of  the  coil,  and  is  usually  represented  by  the  capital  letter  L. 
The  number  of  lines  of  force  passing  through  the  solenoid  when 
the  current  is  one  ampere  is  equal  to 

4 TrnAfj- 
10  l 


Hence  the  value  of  the  self -inductance  of  a long  solenoid  of  uni- 
form ’permeability  may  be  defined  as  times  the  product  of  the 

number  of  turns  in  the  circuit  by  the  number  of  lines  of  force 
inclosed  by  the  circuit  when  carrying  one  ampere  of  current.  That 
is,  in  the  case  of  a long  solenoid  in  a medium  of  uniform  per- 
meability, n 0 

L 108  x i ' 

* Jackson’s  Electro-magnetism  and  the  Construction  of  Dynamos , p.  15. 
t Jackson's  Electro-magnetism  and  the  Construction  of  Dynamos , pp.  6 
and  7. 


136 


ALTERNATING  CURRENTS 


Since  the  number  of  lines  of  force  developed  by  a coil  is  pro- 
portional to  the  number  of  turns  composing  it,  its  self-inductance 
is  proportional  to  the  square  of  its  turns.  Also,  it  will  be  ob- 
served that  the  formula  for  self-induced  electro-motive  force 
reduces  to  7. 

t -di 

“ = ~ Lit 


42.  The  Henry.  — The  Chicago  Electrical  Congress  formally 
assigned  the  name  Henry  to  the  unit  in  which  self-inductance 
is  measured,  after  Professor  Joseph  Henry,  the  notable  Ameri- 
can discoverer  of  electro-magnetic  phenomena.  Since  the  volt 
is  108  times  as  large  as  a C.  G.  S.  unit  of  electro-motive  force  and 
the  ampere  is  as  large  as  the  C.  G.  S.  unit  of  current,  it  is 
apparent  from  the  formula  that  the  Henry  is  109  times  as  large 
as  the  corresponding  C.  G.  S.  unit  of  self-inductance. 

43.  Self-inductance  of  a Short  Coil ; Circuit  containing  Variable 
Permeability  ; Examples.  — The  definition  of  the  henry  is  above 

developed  for  a long  solenoid 
in  which  all  the  lines  of  force 
pass  through  all  the  turns. 
If  the  circuit  is  not  so  con- 
structed, the  definition  still 
holds,  but  the  summation  of 
the  number  of  lines  of  force 
passing  through  each  turn 
individually  must  be  taken, 
since  the  number  of  lines 
passing  through  any  turn  is 
a variable  which  depends 
upon  the  position  of  the  turn 
in  the  coil.  Thus,  suppose  Fig.  116  represents  a short  solenoid 
of  eight  turns  in  which  are  developed  ten  lines  of  force  when 
one  ampere  flows  through  the  coil.  Assuming  the  distribution 
of  the  lines  shown  in  the  figure,  the  self-inductance  is  calcu- 
lated as  follows : 


Fig.  116.  — Solenoid  and  Magnetic  Field. 


10x2  + 8x2+6x2  + 4x2 
10s 


56 

io8 


henrys. 


If  an  iron  core  is  now  placed  in  the  coil,  the  number  of  lines 
of  force  is  increased  directly  as  the  reluctance  of  the  magnetic 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  137 


circuit  is  decreased. 


Hence,  assuming  the  distribution  of  the 

56  P' 

lines  to  remain  unchanged,  the  self-inductance  becomes 
P' 

henrys,  where  — 


108P 

is  the  ratio  of  the  reluctance  before  and  after 


the  iron  core  is  inserted. 

In  the  case  of  a long  solenoid 


T n 4 7 rnA 
L = — x 


108 


10  Z 


when  the  permeability  is  unity,  but  when  the  permeability  of 
the  magnetic  circuit  taken  as  a whole  is  g,  the  number  of  lines 
of  force  per  ampere,  as  shown  before,  is 


4 7 ruA/i 


101 


and  therefore 


T 11 

i = foSx 


4 TrnAfi 

10  l 


In  general,  where  the  magnetic  circuit  is  composed  wholly  of 
non- magnetic  material,  the  self-inductance  is 

t _ n <f> 

108  i 


(where  </>  represents  the  number  of  lines  of  force  for  current  if 
and  the  inductance  is  a constant  for  all  values  of  i.  However, 
when  iron  or  other  magnetic  material  is  included  in  the  mag- 
netic circuit,  the  value  of  the  self-inductance  varies  with  the 
value  of  i because  g varies  with  fi.  As  before, 


10  H 

hut  this  may  have  a different  value  for  each  value  of  i since 

4 7 rnA/jii 


4>  = 


10  l 


and  /r  varies  with  0.  In  this  case  the  inductance  for  any  value 
of  i is  n times  as  great  as  when  no  magnetic  material  is  in- 
cluded in  the  magnetic  circuit,  the  value  of  fi  taken  being  that 
corresponding  to  the  particular  value  of  i. 

Therefore,  the  self -inductance  of  a long  solenoid  ivliich  contains 
an  iron  core , when  carrying  a certain  current , may  be  defined  as 


138 


A LT E RNATING  C U li  R ENTS 


- times  the  number  of  turns  in  the  solenoid  multiplied  by  the 

number  of  lines  of  force  set  up  by  the  current  and  divided  by  the 
number  of  amperes  of  the  current.  It  is  shown  later  that  the 
self-inductance  found  by  this  relation  is  not  necessarily  equal 
to  the  apparent  self-inductance  found  by  the  use  of  measuring 
instruments  because  the  disturbing  effects  of  eddy  currents  and 
hysteresis  in  iron  cores  sometimes  mask  the  results. 

As  an  example  of  the  calculation  of  the  value  of  A,  consider 
a uniform  ring  of  wrought  iron  100  centimeters  in  mean  cir- 
cumference and  20  square  centimeters  in  cross  section.  Sup- 
pose a coil  of  2500  turns  is  uniformly  wound  on  the  ring,  and  a 
current  of  two  amperes  is  passed  through  the  magnetizing  coil. 
Taking  y as  equal  to  250,  which  is  a fair  value, 


_ 4 7 mAyl _ 
10 1 ~ 


4 7r  x 2500  x 20  x 250  x 2 
10  x 100 


314,200 ; 


hence, 


L = 


n<f>  _ 2500  x 314,200 
108/  108  x 2 


= 3.93  henrys. 


If  the  current  in  the  magnetizing  coil  is  taken  as  1|,  and  it  is 
supposed  that  the  value  of  y becomes  roughly  300,  then 


T 2500  x 282,744  , 71  , 

108  x 1.5  J 


That  is,  the  permeability  having  increased  with  the  reduction 
of  the  magnetic  density  in  the  iron  core,  the  magnetism  per 
ampere  is  greater  with  the  smaller  current  and  the  self-induc- 
tance is  therefore  larger,  although  the  total  magnetism  is  smaller. 

If  a similar  winding  is  put  on  a ring  of  the  same  dimensions 
but  of  brass  or  other  non-magnetic  material,  the  magnetism  set 
up  per  ampere  is: 


4>  = 


4 7T  x 2500  x 20  x 1 
10  x 100 


628.3, 


and  the  self-inductance  is  : 


L = 


2500x628.3 
108  x 1 


= .0157  henry. 


In  this  case  the  self-inductance  is  not  affected  by  the  strength 
of  the  current,  because  the  permeability  of  the  medium  is  fixed 
and  independent  of  the  current  flow. 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  139 


According  to  the  foregoing  definitions,  the  number  of  henrys 
of  self-inductance  possessed  by  any  circuit,  whatever  may  be 

the  current  flow,  is  equal  to  -i-  times  the  change  of  the  sum- 


mation of  the  linkages  of  lines  of  flux  through  or  around  the  con- 
ductors of  the  circuit  which  would  he  caused  by  a change  of  the 
current  by  one  ampere,  assuming  that  the  magnetic  reluctance 
of  the  surrounding  medium  does  not  change  at  the  same  time. 
In  the  case  of  the  long  solenoid  or  the  ring  solenoid,  all  of  the 
lines  of  flux  link  through  the  circuit  as  many  times  as  there 
are  turns  in  the  coil,  and  the  summation  of  linkages  may  be 
found  by  multiplying  the  total  number  of  lines  of  flux  by  all 
of  the  turns  of  the  coil ; hut  in  other  instances  some  of  the 
lines  of  flux  make  a different  number  of  linkages  with  the  cir- 
cuit from  others,  and  then  the  summation  of  the  linkages  can 
be  determined  only  by  tracing  the  linkages  of  the  lines  indi- 
vidually and  summing  them  up  as  in  the  example  on  page  136. 
The  self-inductance  ordinarily  required  in  dealing  with  alter- 
nating-current circuits,  is  that  calculated  upon  the  basis  of  the 
circuital  magnetic  permeability  obtaining  for  the  maximum  in- 
stantaneous ordinate  of  the  current  under  consideration. 

In  the  usual  practical  problems  that  are  met  by  the  engineer, 
the  conformation  and  numerical  constants  of  the  magnetic  cir- 
cuits and  their  windings  are  often  unknown  or  are  so  irregular 
that  the  self-inductance  cannot  be  determined  by  calculation, 
and  experimental  determination  must  then  be  resorted  to  ; but 
the  same  definitions  apply  to  the  units  and  the  same  physical 
conceptions  may  be  cultivated. 


The  formula  et  = — L 


di 

dt 


obviously  only  applies  to  conditions 


in  which  L is  constant,  which  is  not  the  case  when  the  magnetic 
circuit  includes  an  iron  core.  In  the  latter  case,  the  formula 
becomes  : 


d(Li) 


et=- 


dt 


_adJjn)  = _a 
dt 


di  , -da 
a 1-  l — 

dt  dt 


If  the  winding  under  consideration  is  a long  solenoid  on  an 
iron  core,  the  constant  a in  this  equation  is  expressed  by 

n 4 7 rnA 

a = — x . 

108  10 1 


140 


ALTERNATING  CURRENTS 


It  will  be  observed  that  the  first  term  of  the  last  member  of  the 

foregoing  equation  for  et  is  equal  to  Z . The  second  term  of 

the  same  member  shows  the  effect  on  the  induced  voltage  which 
is  produced  by  the  change  of  the  permeability  of  the  iron  in  the 
magnetic  circuit  as  the  current  and  magnetic  density  change. 
The  algebraic  sign  of  the  first  term  is  fixed  by  the  direction 

of  change  of  the  current,  ^ ; but  the  second  term  may  be 

at 

either  positive  or  negative  with  respect  to  the  algebraic  sign  of 
depending  on  which  limb  of  the  permeability  curve  comes  into 


the  conditions  concerned,  — that  is,  depending  upon  whether 
the  permeability  increases  or  decreases  as  the  current  rises. 

The  induced  voltage  et  is  therefore  numerically  equal  to 

L y only  when  L is  constant,  and  it  may  be  numerically  either 
larger  or  smaller  than  L -T  when  the  permeability  of  the  mag- 


netic circuit  concerned  varies  as  the  magnetic  density  and  cur- 
rent change.  It  is  numerically  larger  than  L ^ when  the  per- 


meability varies  similarly  as  the  magnetic  density  and  current 
(i.e.  increases  when  they  increase  and  decreases  when  they 

decrease);  and  it  is  numerically  smaller  than  Z~  when  the 

permeability  varies  inversely  to  the  magnetic  density  and  cur- 
rent, as  it  does  when  the  magnetic  density  is  larger  than  that 
producing  the  maximum  permeability  in  the  iron. 


Prob.  1.  A closed  ring  of  iron  contains  a flux  of  2,000,000 
lines  when  a current  of  5 amperes  flows  through  its  magnetizing 
coil  of  1500  turns.  What  is  the  self-inductance  of  the  coil  ? 
In  this  and  the  following  problems  the  effect  of  magnetic  leak- 
age is  assumed  to  be  of  negligible  magnitude. 

Prob.  2.  A closed  ring  having  a constant  permeability  of 
1000  units  and  a cross  section  of  200  square  centimeters  and 
a length  of  150  centimeters,  has  wound  upon  it  a coil  con- 
taining 2000  turns  of  wire.  What  is  the  self-inductance  of 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  141 


M 

the  coil?  (Aid:  remember  that  the  flux  <E>  = — , where  M is 

4 7 nil 


10 


magneto-motive  force  and  P reluctance ; that  M = 

and  P = where  l is  the  length  of  the  magnetic  circuit  and 
A/x 

A is  its  cross  section.) 

Prob.  3.  In  problem  2 the  ring  is  cut  and  its  ends  pulled 
apart  until  there  is  an  air  space  included  in  the  magnetic  circuit 
having  a length  of  a half  centimeter  and  an  effective  cross 
section  of  280  square  centimeters.  What  is  the  self-inductance 
of  the  coil,  supposing  that  the  permeability  of  the  iron  has  not 
changed  ? 


Prob.  4.  A ring  of  iron  wire  100  centimeters  long  and  400 
square  centimeters  in  cross  section  is  surrounded  by  a magnet- 
izing coil  of  200  turns.  When  the  magnetic  density  in  the 
iron  is  2000  lines  of  force  per  square  centimeter,  /x  = 1000 ; 
when  the  magnetic  density  is  5000,  /x  = 1200;  when  the  mag- 
netic density  is  10,000,  /x  = 800  ; when  the  magnetic  density  is 
20,000,  /x  = 300 ; when  the  magnetic  density  is  40,000,  /x  = 50. 
What  is  the  self-inductance  of  the  coil  for  each  value  of  the 
five  magnetizations  ? 


Prob.  5.  The  field  windings  of  a bi-polar  direct-current  dy- 
namo contain  2000  turns  carrying  a current  of  2|  amperes,  the 
field  magnet  has  an  average  length  of  150  centimeters,  an  aver- 
age cross  section  of  500  square  centimeters,  and  an  average  per- 
meability of  800;  the  armature  has  the  same  average  magnetic 
cross  section  and  a length  of  30  centimeters  with  a permeability 
of  1500;  and  the  two  air  spaces  each  have  a length  of  1 centi- 
meter and  a cross  section  of  600  square  centimeters.  What  is 
the  self-inductance  of  the  exciting  winding  ? 


44.  Examples  of  Self-inductances.  — Ordinary  practical  experi- 
ence in  electrical  measurements  and  in  handling  wires  soon  gives 
a capacity  for  estimating  the  values  of  resistances ; in  the  same 
way  facility  is  soon  gained  in  roughly  estimating  electrostatic 
capacities,  or  the  current  which  maybe  safely  carried  by  a wire, 
or  even  the  ampere  turns  required  to  produce  a given  magnet- 
ization in  a magnetic  circuit.  Ordinary  practice,  however, 
gives  little  clue  to  estimating  the  self-inductance  in  a circuit. 


142 


ALTERNATING  CURRENTS 


It  is  true  that,  as  already  shown,  the  self-inductance  is  depend- 
ent upon  the  magnetism  inclosed  in  the  circuit  and  the  number 
of  turns  thereof,  but  experience  in  dealing  with  coils  and  mag- 
netic circuits  is  not  usually  regarded  in  such  a way  as  to  aid 
in  estimating  self-inductances.  The  following  values  of  self- 
inductance are  therefore  presented  here  to  give  a foundation  for 
judgment. 

The  range  of  self-inductances  met  in  practice  is  very  wride. 
The  smallest  which  are  practically  met  are  in  the  doubly 
wound  resistance  coils  used  for  Wheatstone  bridges  and  similar 
devices.  Since  the  wire  in  these  is  doubled  back  upon  itself, 
the  magnetic  effect  of  the  current  is  almost  neutral  and  the 
inductance  is  often  less  than  a microhenry  (one  millionth  of  a 
henry).  The  inductance  of  a certain  electric  call  bell  of  2.5 
ohms  resistance  has  been  found  to  be  12  microhenrys ; a tele- 
phone call  bell  of  80  ohms  resistance,  1.4  henrys;  a modern 
short-core  telephone  bell  of  1000  ohms  resistance,  5.5  henrys: 
the  armature  of  a magneto  calling  generator  of  550  ohms  resist- 
ance, from  2.7  henrys  when  the  plane  of  the  coil  lay  in  the  plane 
of  the  pole  pieces,  to  7.3  henrys  when  the  plane  of  the  coil  was 
perpendicular  to  the  plane  of  the  pole  pieces ; more  modern 
magneto  generator  armatures,  one  measuring  540  ohms  resist- 
ance, 1.6  henrys  with  plane  of  coil  in  plane  of  pole  pieces,  2.4 
henrys  with  plane  of  coil  perpendicular  to  plane  of  pole  pieces: 
one  measuring  125  ohms  resistance,  .51  and  .74  henry;  and 
one  measuring  110  ohms  resistance,  .89  and  1.17  henrys  ; a Bell 
telephone  receiver  measuring  75  ohms  resistance,  with  dia- 
phragm, 75  to  100  millihenrys  (thousandths  of  henrys),  without 
the  diaphragm  about  35  per  cent  less;  a modern  bipolar  telephone 
receiver  with  70  ohms  resistance,  35  millihenrys;  mirror  gal- 
vanometers vary  with  their  resistance  from  a few  millihenrys 
to  10  or  12  henrys ; a mirror  galvanometer  for  submarine  sig- 
naling of  2250  ohms  resistance,  3.6  henrys;  astatic  min'or 
galvanometers  of  5000  ohms  resistance  average  about  2 henrys. 
The  single  coil  of  a Thomson  galvanometer  of  2700  ohms 
resistance  measured  2.56  henrys;  the  coil  of  another  Thomson 
galvanometer  having  100,000  ohms  resistance  measured  70 
henrys  ; the  coil  of  an  Ayrton  and  Perry  spring  voltmeter,  with- 
out iron  core,  measured  1.462  henrys.  This  coil  had  a length 
of  2.88  inches,  an  external  diameter  of  3 inches,  was  wound  on 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  143 


a brass  tube  .58  inch  in  external  diameter,  and  had  a resistance 
of  338.5  ohms.  Each  of  the  above  measurements  was  made 
with  a current  of  a few  milliamperes.  The  following  are  meas- 
urements of  telegraphic  apparatus: 


TABLE 

Polarized  Relays  of  Various  Types 


Type 


1 

2 

3 

4 


P.esistance  in  Ohms 


419 

423 

413 

413 


Self-inductance  in 
Henrys 


1.99 

1.89 

1.69 

1.31 


Testing  Current  in 
Milliamperes 


6.3 

6.3 

6.3 

6.3 


All  armatures  were  4 mils  (thousandths  of  an  inch)  from  the 
magnet  poles. 

A common  Morse  relay  of  148  ohms  resistance  measured 
10.47  henrys  with  the  armature  against  the  poles,  and  3.71 
henrys  witli  the  armature  20  mils  from  the  poles,  the  measur- 
ing current  being  6.3  milliamperes.  In  ordinary  working  ad- 
justment the  inductance  of  a Morse  relay  is  about  5 henrys. 
Telegraph  sounders  with  bobbins,  respectively,  1|  by  1,  and  1| 
by  1^  inches,  each  wound  to  20  ohms  resistance,  measured  191 
and  150  millihenrys,  the  armatures  being  4 mils  from  the  poles 
and  the  measuring  current  being  125  milliamperes.  A single 
coil  of  a Morse  sounder  with  a resistance  of  32  ohms,  and  hav- 
ing an  iron  core  .31  inch  in  diameter  and  3 inches  long,  the 
bobbin  being  .94  inch  in  diameter,  was  found  to  have  a self- 
inductance of  94  millihenrys.  A complete  sounder  with  a core 
like  that  of  the  preceding  coil,  but  with  bobbins  of  50  ohms 
resistance  having  a diameter  of  1.25  inches,  was  found  to  have 
a self-inductance  of  444  millihenrys.  The  self-inductance  of  a 
complete  sounder  of  14  ohms  resistance  measured  265  milli- 
henrys ; a modern  telephone  relay  with  62  ohms  resistance,  300 
millihenrys ; another  with  2000  ohms  resistance,  840  milli- 
henrys ; a supervisory  relay  for  telephone  service  measuring 
12.5  ohms,  approximately  1 millihenry;  a modern  telephone 
induction  coil,  primary  coil  resistance  17  ohms,  self-inductance 
117  millihenrys,  secondary  coil  resistance  27  ohms,  self-induc- 


144 


ALTERNATING  CURRENTS 


tance  74  millihenrys,  mutual  inductance  approximately  91 
millihenrys ; a modern  telephone  repeating  coil  with  46  ohms 
resistance  in  each  coil,  1.10  henrys  self-inductance  in  each  coil 
and  1.09  henrys  mutual  inductance.  These  measurements  of 
telephone  apparatus  were  made  with  alternating  current  of  a 
frequency  of  800  periods  per  second. 

A single  phase  transmission  line  of  No.  2 B.  and  S.  gauge 
copper  wires  spaced  36  inches  apart  is  calculated  to  have  a 
resistance  of  about  1.62  ohms  and  a self-inductance  of  about 
3.78  millihenrys  per  mile;  No.  6 copper  wire  under  similar 
conditions  to  have  a resistance  of  about  4.09  ohms  and  a self- 
inductance of  about  4.08  millihenrys  per  mile. 

Bare  No.  12  B.  and  S.  gauge  copper  wire  erected  on  a pole 
line  about  23  feet  from  the  ground  is  calculated  by  Kennedy 
to  measure  about  8.5  ohms  and  3.15  millihenrys  per  mile; 
No.  6 copper  wire  under  similar  conditions  is  calculated  to 
measure  about  2.1  ohms  and  2.95  millihenrys.  A quadruplex 
telegraph  line,  with  all  instruments  in  circuit,  measures  approxi- 
mately 10  henrys. 

The  largest  self-inductances  met  in  practice  are  usually  in 
the  windings  of  induction  coils  or  of  electrical  machinery. 
The  secondary  winding  of  an  induction  coil  capable  of  giving 
a 2-inch  spark  and  having  a resistance  of  5700  ohms,  meas- 
ured 51.2  henrys.  The  primary  winding  of  an  induction  coil 
which  is  19  inches  long  and  8 inches  in  diameter,  measured  .145 
ohm  and  13  millihenrys,  while  its  secondary  measured  30,600 
ohms  and  2000  henrys.  The  inductance  of  dynamo  fields  is 
likely  to  vary  from  1 to  1000  henrys ; direct-current  dynamo 
armatures  measure  between  the  brushes  from  .02  to  50  henrys ; 
the  fields  of  a shunt-wound  Mather  and  Platt  direct-current 
dynamo  built  for  an  output  of  100  volts  and  35  amperes  meas- 
ured 44  ohms  and  13.6  henrys  at  a small  excitation  ; the  arma- 
ture of  the  same  machine  measured  .215  ohm  and  .005  henry; 
a Mordey  alternator  armature  of  the  disk  type,  with  a capacity 
for  18  amperes  at  2000  volts,  measured  2 ohms  and  .035  henry ; 
a Kapp  alternator  armature  of  the  ring  type,  with  a capacity 
of  60  kilowatts  at  2000  volts  measured  1.94  ohms  and  .069 
henry ; another  Kapp  machine,  30  kilowatts,  2000  volts,  meas- 
ured 7 ohms  and  .0977  henry;  the  fields  of  a Ferranti  alterna- 
tor measured  3 ohms  and  .61  henry,  while  the  armature  of  the 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  145 

same  machine  built  for  an  output  of  200  volts  and  40  amperes 
measured  .0011  to  .0013  henry,  with  no  current  in  fields  ; the 
armature  of  a British  Thomson-Houston  alternator,  three-phase, 
850  kilowatts,  5000  volts,  measured  .332  ohm  in  resistance 
and  .061  henry  in  self-inductance  per  phase ; the  armature  of  a 
General  Electric  alternator,  three-phase,  2500  kilowatts,  6500 
volts,  measured  .41  ohm  in  resistance  and  .035  henry  in  self- 
inductance per  phase ; while  another  British  Thomson-Hous- 
ton machine,  a turbo-alternator,  1500  kilowatts,  1000  volts,  gave 
an  inductance  per  phase  of  .0403  henry ; the  primary  and 
secondary  windings  of  transformers  measure  roughly  from  .001 
of  a henry  up  to  50  henrys,  depending  upon  their  output  and 
the  voltage  for  which  they  are  designed. 

An  electro-magnet  designed  by  H.  DuBois  * had,  when 
carrying  a current  of  45  amperes,  a self-inductance  of  180 
henrys.  This  electro-magnet  had  a core  of  iron  made  of  two 
half  rings  butted  together,  which  had  a radius  of  25  centi- 
meters and  a cross  section  of  78.5  square  centimeters.  The 
core  was  wound  with  12  coils  of  200  turns  each.  The  self- 
inductance given  is  for  all  coils  in  series. 

The  effect  of  the  field  magnets  upon  the  self-inductance  of 
a disk-alternator  armature  is  shown  by  some  measurements 
taken  by  Dr.  Duncan  f on  a small  Siemens  eight-pole  alterna- 
tor, the  results  of  which  are  given  in  the  following  table. 


TABLE 

Self-inductance  of  Armature  in  Place 


Position  of  Armatuke 

Current  in  Field 

0° 

111° 

22j° 

0 ampere 

.120 

.112 

.100 

2.5  amperes 

.115 

.108 

4.5  amperes 

.128 

.115 

.106 

Self-inductance  of  armature  removed  from  field,  .082  henry; 
resistance  of  armature,  7 ohms ; pitch  of  the  poles,  45°. 

Professor  Ayrton  found  that  the  self-inductance  of  an  unex- 

* The  Magnetic  Circuit,  H.  DuBois,  p.  264. 
t Electrical  World,  Vol.  11,  p.  212. 

L 


146 


ALTERNATING  CURRENTS 


cited  Mordey  alternator  armature  varied  between  .033  and 
. 038  henry,  and  that  this  decreased  about  10  per  cent  when  the 
fields  were  excited.* 

45.  Conditions  of  Establishment  and  Termination  of  Current 
in  a Circuit  containing  Resistance  and  Self-inductance  in  Series.  — 

a.  Rise  of  Current  under  Constant  Impressed  Voltage.  — If  a 
circuit  containing  constant  self-inductance  and  resistance  is 
suddenly  connected  to  a source  of  constant  voltage,  the  current 
is  retarded  so  that  its  rise  is  along  a logarithmic  curve.  The 
voltage  E of  the  source  is  absorbed  at  each  instant  in  sup- 
plying the  iR  drop  and  overcoming  the  counter-voltage 


which  represents  the  instantaneous  value  of  the  current  flowing 
at  any  moment  while  the  voltage  E , constant  for  the  time 
under  consideration,  is  applied  to  a circuit  of  constant  induc- 
tance L.  To  find  the  value  of  the  instantaneous  current  at 
any  particular  time  t,  we  have  from  the  same  equation,  by 
transposition, 


— L — • Hence  the  equation 
dt 


and  from  it  is  given 


Ldi 

dt 


di  dt 


E - iR  L ; 


whence 


which  gives 


oi- 


and  finally 


* Jour.  Inst.  Elect.  Eng.,  Yol.  18,  p.  662;  also  ibid.,  p.  654. 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  147 


Fig.  117.  — Logarithmic  Curve  of  Rising  Current. 


where  e is  the  base 
of  the  Naperian  sys- 
tem of  logarithms. 
Figure  117  shows 
the  logarithmic 
curve  of  rise  of  cur- 
rent in  a particular 
circuit  and  Fig. 
117  a shows  the 


di  E 

rate  of  change  — = — e r of  the  same  current. 
dt  L 

It  will  be  observed  that,  since  in  this  case  the  counter-voltage 
Ldi 

of  self-inductance, — , is  equal  to  — QE  — iK),  its  instan- 

ce 


taneous  value  is  represented  by  the  exponential  expression 

_Rt 

— Ee^,  where  t is  the  numerical  value  of  the  time  for  the 

di 

instant  at  which  the  ratio  — is  taken. 

dt 

b.  Fall  of  Current  on  Withdrawal  of  Impressed  Voltage.  — 
Likewise,  when  the  voltage  is  withdrawn,  E = 0,  and  if  the 


or 


- (log  i-  log  7) 


t_ 

r 


i i z Rt 

and  logj=-— , 

R 

in  which  I is  the  value  of  the  current  equal  to  -— . 


9 


. R 

l = Re~L' 


Hence, 


148 


ALTERNATING  CURRENTS 


which  gives  the  instantaneous  value  of  the  current  at  any 
instant  during  its  fall,  after  the  voltage  is  withdrawn.  In  this 
Ldi 

is  the  voltage  of  self-induction  causing  the  current 


case  — 


dt 


to  flow,  and  it  is  obviously  equal  at  each  instant  to  iR  — EV~l. 

Figure  118  shows 
the  logarithmic  curve 
of  fall  of  current  for 
the  circuit  already 
referred  to  by  Fig. 
117,  and  Fig.  118  a 
shows  the  rate  of 
change  of  the  falling 
current. 

c.  Current  Value 
when  the  Impressed 
— Under  these  condi- 


Fig.  118.  — Logarithmic  Curve  of  Falling  Current. 


Voltage  is  a Sine  Function  of  the  Time 
tions  the  voltage  equation  of  the  circuit  becomes 

Ldi 
dt 


e =f(t)  = em  sin  cot=  Ri  + 


In  this  expression  a>  = 2 irf  and  cot  — a.  This  is  a linear  differ- 
ential equation  of  the  first  order,  the  solution  of  which  is  * 


i — 


V7U  + 4 


sin  («  — d)  + 


in  which 


6 = tan-1 


2 irfL 
R 


After  a short  time  the 
exponential  term  in  the 
right-hand  member  of  the 
equation  becomes  inap- 
preciable since  the  expo- 
nent is  negative  and  t is 
an  increasing  quantity. 
It  may  be  neglected  for 
the  present.  Its  effect 
will  be  discussed  later,  f 
The  current  now  becomes 
a sine  function  of  the  time 


Fig.  118  a . — Logarithmic  Curve,  showing  Rate  of 
Change  of  Falling  Current. 


* Murray’s  Differential  Equations,  p.  26. 


t Art.  59. 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  14S 


and  lags  behind  the  impressed  voltage  by  an  angle  6 whose  tarn 

, . 2 7 rfL 
gent  is 


R 


It  has  the  same  frequency  as  the  voltage. 


It  will  be  observed  that  the  sinusoidal  current  has  a maximum 
value  of 

2 _ ® m 

VW+  4 7T2/2X2’ 

and  since  the  effective  values  bear  a fixed  ratio  to  the  maximum 
values,  we  have 

~ V^  + 4tt2/2L2’ 

in  which  I is  the  effective  value  of  the  current  and  R the 
effective  value  of  the  sinusoidal  impressed  voltage.  The  de- 
nominator of  this  expression  is  measured  in  ohms  and  is  called 
the  Impedance  of  the  alternating-current  circuit.  This  im- 
pedance is  composed  of  the  square  root  of  the  sum  of  two 
terms.  The  first  of  these  terms  is  the  square  of  the  electrical 
resistance,  and  the  second  of  the  terms  is  the  square  of  the 
expression  2 irfL.  This  latter  expression  is  called  the  Reactance 
of  the  circuit.  The  vector  relations  of  the  voltages  of  a circuit 
containing  resistance  and  self-inductance  are  shown  in  Fig. 
119,  which  corresponds  to  Fig.  115. 

46.  Transference  of  Electricity  during  the  Transient  State. — 
When  the  current  in  an  inductive  circuit  has  reached  its  full 
value,  a smaller  quantity  of 
electricity  has  passed 
through  the  circuit  during 
the  interval  since  the  start 
of  the  current  than  would 
have  passed  if  the  retarda- 
tion, or  momentum  effect, 
had  not  been  present.  This 
decrease  in  the  quantity  of 

electricity  is  proportional  to  the  area  OYQ  between  the  curve 
of  the  current  and  the  horizontal  line  YQ  (Fig.  117).  The 
ordinates  of  the  curve  representing  instantaneous  current 
strengths  in  amperes  and  the  abscissas  representing  time  in 
seconds,  the  area  referred  to  obviously  represents  quantity 
of  electricity  measured  in  coulombs.  This  area  may  be  found 
in  terms  of  Z,  Z,  and  R as  follows : 


Er  =IR 

Fig.  119. — Voltage  Relations  in  Circuit  con- 
taining Resistance  and  Inductance. 


150 


ALTERNATING  CURRENTS 


The  area  is  equal  to  J'  — i) dt,  where  T is  the  duration 


of  time  from  the  instant  of  the  introduction  of  the  source  of 
voltage,  E,  into  the  circuit  to  the  instant  at  which  the  current 

E 

reaches  its  full  or  ultimate  value  /=  — • Also,  since  iR  = 

R 

E — L—  and  * = — — it  is  manifest  that 
dt  R Rdt 


.r(i->=x 


’Ldi 

~R 

E , 


LI 
R ‘ 


The  area  OYQ  A,  which  is  equal  to  — T,  represents  the  number 

R 

of  coulombs  which  would  have  been  transferred  in  the  circuit 
during  time  T had  the  current  instantly  come  to  its  full  or 
E 

ultimate  value  of  1=  A;  and  the  area  OQA , which  is  equal  to 
ET  LI 

— — — , represents  the  number  of  coulombs  that  are  actually 

transferred  through  the  inductive  circuit  while  the  current  is 
rising  over  the  logarithmic  curve  to  its  full  value.  The  quan- 
tity coulombs  is  equal  to  the  difference  between  the  quan- 
tity of  electricity  which  actually  flows  through  the  circuit  in 
the  period  during  which  the  current  is  rising  to  its  permanent 
E 

value  I—  — and  the  quantity  that  would  flow  in  the  same  time 
R 

if  the  current  immediately  rose  to  its  full  value. 

If  the  voltage  is  suddenly  reduced  from  E to  zero  without 
breaking  the  circuit,  the  current  does  not  stop  immediatelj*, 
but  falls  off  along  a logarithmic  curve,  and  the  quantity  of 
electricity  passing  through  the  circuit  is  increased  on  this 
account.  The  increased  quantity  is  proportional  to  the  area 
OYQ  in  Fig.  118,  which  shows  a curve  of  falling  current  in  an 
inductive  circuit  corresponding  with  the  curve  of  rising  cur- 
rent shown  in  Fig.  117.  That  this  quantity,  proportional  to 
area  OYQ  of  Fig.  118,  which  is  transferred  through  the  cir- 
cuit after  the  voltage  E is  removed,  is  equal  to  the  deficit  of 
electricity  during  the  starting  of  the  current  in  the  same  cir- 

di 

cuit  is  shown  thus:  the  counter-voltage  is,  as  before,  — Z — 

dt 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  151 


and  = 

R 


Ldi 

Rdt 


But  the  area  OYQ  of  Fig.  118  is  equal  to 


dt 


-Jf 


0 Ldi 
R 


LI 
R ’ 


where  T is  the  time  in  seconds  measured  from  the  instant  of 
removing  the  source  of  voltage  E to  the  instant  at  which  the 

current  reaches  zero  value,  and  I is  the  value  ^ of  the  current 

R 

at  the  instant  of  removing  the  source  of  voltage. 

The  current,  which  is  thus  maintained  by  the  disappearing 
magnetic  field  after  the  external  source  of  current  has  been 


removed  and  which  conveys  the  quantity  of  electricity 


LI 

R 


coulombs,  was  formerly  called  the  extra  current  of  self-induc- 
tion. It  was  then  thought  that  the  self-induction  caused  an 
increase  of  the  flow  of  electricity  through  the  circuit ; but  the 
foregoing  demonstration  of  the  manner  in  which  the  current 
rises  and  falls  in  a self-inductive  circuit  shows  that  the  extra 
current , which  occurs  when  the  external  source  of  current  has 
been  removed  from  the  circuit  and  the  current  falls  from  I to 


0,  conveys  a quantity  of  electricity, 


which  is  exactly 


equal  to  the  deficit  of  coulombs  which  was  caused  at  the  time 
that  current  was  introduced  into  the  circuit,  — the  aforesaid 
deficit  being  due  to  the  fact  that  the  self-induction  causes  the 
current  to  rise  gradually  instead  of  instantly  when  the  source 
of  voltage  is  switched  into  the  circuit,  and  the  extra  current 
being  due  to  the  fact  that  the  self-induction  maintains  the  cur- 
rent while  the  magnetic  field  is  dying  away  after  the  external 
source  of  voltage  is  removed.  If  the  value  I'  is  substituted 
for  the  value  0 in  the  integrations,  it  will  be  observed  that  the 
deficit  coulombs  upon  the  current  being  increased  from  I'  to  1 
and  the  extra-current  coulombs  upon  the  restoration  of  the 
current  to  I\  assuming  these  operations  accomplished  without 


changing  the  circuit  constants,  are  each  equal  to 


LCL-T) 


and  the  principle  may  be  stated  as  follows : 

If  a current  is  started  or  changed  by  changing  the  voltage 
in  a circuit  having  only  resistance  and  self-inductance,  and 
these  of  fixed  values,  and  the  voltage  is  then  restored  to  its 


152 


ALTERNATING  CURRENTS 


original  value  and  the  current  allowed  to  return  to  its  original 
or  initial  value  without  changing  the  resistance  or  self-induc- 
tance of  the  circuit,  the  total  number  of  coulombs  transferred 
through  the  circuit  during  the  cycle  are  equal  to  the  number 
that  would  be  transferred  if  the  circuit  were  without  self- 
inductance. 

The  foregoing  demonstration  is  founded  on  the  proposition 
that  the  self-inductance  of  the  circuit  is  fixed  in  value,  and  that 
the  magnetic  linkages  existing  at  the  respective  ends  of  a cycle 
of  changes  are  proportional  to  the  initial  and  final  values  of  the 
current ; but  if  the  electric  circuit  has  an  iron  core,  the  density 
of  the  magnetism  may  not  fall  to  its  initial  value  upon  restoring 
the  current,  and  the  energy  restored  is  not  then  equal  to  that 
absorbed  in  building  up  the  magnetic  flux.  The  difference  in 
the  energy  remains  stored  in  the  magnetic  flux  in  the  form  of  mag- 
netic linkages  maintained  by  residual  magnetism.  Under  the 
circumstances  here  referred  to,  the  permeability  of  the  mag- 
netic circuit  usually  varies  as  some  complex  function  of  the  cur- 
rent, and  the  current  therefore  does  not  rise  and  fall  in  plain 
logarithmic  curves  because  the  value  of  L in  the  equation 

ixdt  — is  a function  of  i of  greater  or  less  complexity.  In 

R 

this  case  the  integration  along  the  curves  of  rise  and  fall  may 
not  be  readily  accomplished,  but  it  always  remains  a fact  (pro- 
vided the  circuit  is  affected  only  by  resistance  and  self-induc- 
tance and  the  resistance  is  of  fixed  value)  that  the  coulombs 
conveyed  through  the  circuit  by  the  extra  current  upon  restor- 
ing a current  to  its  initial  value  are  equal  to  the  deficit  occur- 
ring during  the  rise  of  the  current,  provided  the  magnetism 
and  the  current  both  return  to  their  initial  values.  The  effect 
of  hysteresis  must  be  negligible  in  order  that  this  condition 
may  occur. 

In  case  a current  is  started  by  switching  a source  of  voltage 
into  a circuit  surrounding  an  iron  core  which  is  already  magnet- 
ized, the  current  will  rise  more  rapidly  than  if  the  core  were  not 
previously  magnetized,  if  the  magnetic  force  of  the  current  is 
in  the  direction  of  the  already  existing  magnetism;  and  the 
rise  of  the  current  will  be  retarded  if  its  magnetizing  effect  is 
opposite  to  the  already  existing  magnetism.  This  is  an  obvious 
corollary  from  the  foregoing  demonstrations. 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  153 


Prob.  1.  A steady  current  of  10  amperes  is  flowing  through 
a coil  having  a self-inductance  of  100  henrys  and  a resistance 
of  10  ohms.  How  many  coulombs  of  electricity  will  be  trans- 
ferred through  the  circuit  after  the  voltage  is  withdrawn  with- 
out breaking  the  circuit  or  changing  its  resistance?  Assume 
magnetic  leakage  and  hysteresis  to  be  negligible  in  this  and 
the  following  problems. 

Prob.  2.  A magnetic  circuit  in  the  form  of  a ring  of  200 
centimeters  mean  length  and  100  square  centimeters  in  cross 
section,  with  a fixed  permeability  of  1000  units,  has  a coil 
wound  upon  it  of  1000  turns  and  20  ohms  resistance.  If  a steady 
current  of  10  amperes  is  flowing  in  the  coil,  what  will  be  the 
number  of  coulombs  of  electricity  which  will  be  transferred 
through  the  circuit  after  the  voltage  is  withdrawn  without 
breaking  the  circuit  or  changing  its  resistance  ? 

Prob.  3.  Draw  the  curves  of  quantity  and  current  when  the 
voltage  is  introduced  and  withdrawn  in  the  circuits  in  problems 
1 and  2,  assuming  no  change  of  resistance  or  self-inductance  in 
either  instance. 

Prob.  4.  What  is  the  value  in  amperes  of  the  current  flow- 
ing in  the  circuit  of  problem  2,  at  an  instant  ^ seconds  after 
the  removal  of  the  voltage? 

47.  Energy  stored  in  a Magnetic  Field  associated  with  an 
Electric  Circuit. — The  effect  of  self-inductance  is  manifested, 
as  already  explained,  by  a counter-voltage  which  tends  to 
retard  a rising  current  and  to  sustain  or  continue  a falling 
current.  That  is,  when  a current  is  introduced  into  a circuit, 
the  inductive  effect  of  the  rising  magnetic  flux  created  by  the 
rising  current  prevents  the  current  from  immediately  coming 
to  a value  equal  to  the  impressed  voltage  divided  by  the 
resistance  of  the  circuit.  Likewise,  when  the  voltage  is  with- 
drawn from  the  circuit,  the  dying  out  of  the  magnetic  flux 
prevents,  by  its  inductive  effect,  the  current  from  immediately 
disappearing.  The  inductive  effect  of  the  magnetic  flux  is 
directly  proportional  to  the  rate  of  change  of  the  summation  of 
the  number  of  linkages  around  the  conductors  of  the  circuit 
that  are  made  by  the  lines  of  force  of  the  magnetic  flux. 
Each  such  magnetic  linkage  set  up  in  connection  with  a cur- 


154 


ALTERNATING  CURRENTS 


rent  in  a circuit  represents  a definite  amount  of  stored  energy ; 
and  the  phenomena  relating  to  the  setting  up  of  the  magnetic 
field  (that  is,  these  linkages)  about  a circuit,  thereby  storing 
energy  by  introducing  a current  in  the  circuit,  and  the  phenom- 
ena relating  to  collapsing  the  linkages  and  recovering  the 
energy  by  withdrawing  the  current,  have  close  analogies  to  the 
phenomena  relating  to  the  energy  of  moving  bodies. 

If  a mechanical  force  is  applied  to  a movable  mass  or  body, 
the  velocity  of  the  body  does  not  immediately  rise  to  its  full  or 
ultimate  value,  but  is  prevented  from  doing  so  by  the  inertia 
of  the  mass,  until  an  amount  of  energy  proportional  to  the 
product  of  the  mass  and  the  square  of  the  ultimate  velocity 
has  been  stored  in  the  body.  Likewise,  when  the  impelling 
force  is  removed  from  a moving  body,  it  will  continue  to  move 
until  all  the  energy  that  was  stored  in  bringing  it  up  to  speed 
is  dissipated  in  overcoming  resistance  to  its  movement.  In 

civ 

case  of  tangible  matter  — — , Mv,  and  M — are  respectively 


the  energy,  momentum,  and  rate  of  change  of  momentum  (that 
is,  counter-force  or  pressure  caused  by  inertia)  of  the  mass  M 
when  moving  at  a velocity  v ; while  in  the  case  of  the  electric 
Li  2 di 

circuit,  — — , Li,  and  L — , as  will  be  seen  later,  may  be  called 

Jmi  GLT/ 


the  energy,  momentum,  and  rate  of  change  of  momentum 
(counter-electric  voltage)  of  its  magnetic  flux.  In  such  anal- 
ogies, electric  current  has  its  counterpart  in  velocity  or  rate 
of  motion,  voltage  in  mechanical  force  or  pressure,  electric 
resistance  in  mechanical  frictional  resistance,  and  self-inductance 
in  mass  or  inertia. 

When  a source  of  constant  voltage  is  switched  into  a circuit 
with  self-inductance  and  resistance,  the  current  gradually  rises 

LJ 

to  its  final  value  of  I = — . Also,  a counter-electric  voltage 

R 

of  self-induction  is  produced  in  the  circuit  by  the  effect  of 
the  rising  magnetic  flux  set  up  by  the  current.  This  counter- 
voltage  at  any  instant  may  be  represented  by  the  expression 
di 

— L~,  assuming  the  permeability  of  the  magnetic  circuit  to 


be  constant. 

The  impressed  voltage  contains  a component  equal  and 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  155 

opposite  to  this  counter-voltage  at  each  instant  and  also  sup- 
plies the  iR  drop,  or,  as  has  already  been  pointed  out, 

E=iR  + L-. 

at 

This  equation  manifestly  is  equally  accurate  when  E is  of  con- 
stant value  or  when  it  represents  the  instantaneous  value  of  a 
variable  impressed  voltage,  provided  the  circuit  contains  only 
resistance  and  self-inductance  (that  is,  the  circuit  is  not  subject 
to  the  effects  of  mutual  induction  or  electrostatic  capacity). 
The  rate  at  which  work  is  being  delivered  from  the  external 
source  of  voltage  and  energy  and  expended  in  the  circuit  is, 
in  watts, 

Ei  = i*R  + Li 

dt 

and  the  work  delivered  and  expended  during  a time  interval 
dt  is,  in  joules, 

dw  = Eidt  = PRdt  + Lidi ; 

and  the  joules  delivered  by  the  source  and  expended  in  the 
circuit  during  a period  of  T seconds  in  which  the  current  rises 
from  0 to  I amperes  is 


The  first  term  of  the  right-hand  member  of  this  equation  repre- 
sents the  aggregate  heat  produced  by  the  current  flowing 
through  the  resistance  of  the  wire.  The  second  term  of  the 


right-hand  member  is  equal  to 


LI2 

2 


and  represents  the  work 


done  by  the  external  source  of  voltage  and  energy  in  over- 
coming the  counter-electro-motive  force,  caused  by  the  magnetic 
flux  while  the  current  is  changing.  This  energy  is  stored  in 
the  magnetic  linkages  of  the  flux  with  the  turns  of  the  electric 
circuit  and  is  maintained  as  long  as  the  current  continues  at 
its  value  I. 

As  we  know  the  value  of  current  at  every  instant  during 
its  rise,*  it  is  practicable  to  compute  the  value  of  the  energy, 


* Art.  45,  a. 


156 


ALTERNATING  CURRENTS 


Fig.  120.  — Energy  Stored  in  Magnetic  Linkages  at  Each 
Instant  during  Period  of  Current  Rise. 


W=  | Li 2,  stored  in 
magnetic  linkages 
at  each  instant  dur- 
ing the  rise  of  cur- 
rent. The  curve 
exhibited  in  Fig. 

120  shows  the  value 
of  the  energy  at  each 
instant  correspond- 
ing to  the  conditions 
illustrated  in  Fig. 

117.  It  is  also  prac- 
ticable to  compute  the  rate  of  the  rise  of  energy,  — = Li  — . in 

dt  dt 

the  magnetic  linkages;  and  the  curve  exhibited  in  Fig.  120a 
shows  the  value  of  this  rate  at  each  instant,  corresponding  to 
the  conditions  illustrated  in  Fig.  117. 

The  equation  shows  that  the  energy  delivered  to  the  circuit  by 
a battery,  dynamo,  or  similar  source,  switched  into  a circuit  having 
resistance  and  self-inductance  is  equal  to  the  sum  of  the  energy 
converted  into  heat  by  the  current  flowing  through  the  resist- 
ance of  the  conductors  and  the  energy  stored  in  the  linkages  of 

the  magnetic  flux  with 
the  conductors  of  the 
circuit.  If  the  source 
supplies  a constant  vol- 
tage, the  latter  portion 
of  energy  is  fixed  by 
the  character  of  the 
circuit  and  the  magni- 
tude of  the  voltage, 
and  is  supplied  by 
the  source  while  the 
current  is  rising1  to  its 


E = ICO  VOLTS 
R = 5 OHM8 

L = 0.1  HENRY 


0 .01  ,02  .03  .04  05 

Fig.  120  a. — Rate  of  Change  of  Stored  Energy  at 
Each  Instant  during  Period  of  Current  Rise. 


E 

steady  value  of  — . 


As  long  as  the  current  continues  of  that 


value  the  magnetic  flux  is  maintained  without  further  expen- 
diture of  energy,  and  the  source  then  furnishes  work  equal  to 
that  converted  into  heat  only.  It  will  be  observed  that  these 
deductions  are  founded  on  the  original  premise  that  the  phe* 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  157 


nomena  of  hysteresis,  eddy  currents,  and  other  effects  of  mutual 
induction  and  the  like  are  absent,  and  the  deductions  are  lim- 
ited to  conditions  in  which  the  phenomena  of  steady  resistance 
and  self-inductance  alone  occur. 

di 

From  the  formula  E = iR  + L — it  is  also  to  be  observed  that 

dt 


idt  = — dt  — — di. 
R R 


The  expression  idt  obviously  represents  the  quantity  of  electricity 
(coulombs)  transferred  through  the  circuit  by  the  current  flow 
during  the  period  of  dt  seconds  at  the  instant  of  time  when  the 
current  is  equal  to  i amperes.  If  the  current  came  instantly 


Ij 

to  its  full  value  of  / = — upon  introducing  voltage  R into  the 
R 


circuit,  the  coulombs  transferred  through  the  circuit  in  any 
period  of  T seconds  would  be  IT ; but  the  current  does  not  rise 
instantly  to  its  full  value  in  a circuit  containing  resistance  and 
self-inductance,  on  account  of  the  electro-magnetic  inertia  of 
the  magnetic  linkages.  During  the  time  of  T seconds  which  is 
taken  by  the  current  in  rising  to  its  full  value  of  I amperes, 
the  number  of  coulombs  transferred  is  less  than  IT  because  the 
current  has  had  a smaller  average  value  than  I amperes.  The 
actual  number  of  coulombs  transferred  during  this  transient 
period  of  the  current  rise  may  be  determined  by  integrating 
the  foregoing  formula  from  zero  of  time  and  current  through 
the  elapsed  time  of  T seconds  during  which  the  current  rises  to 
its  full  value  of  I amperes,  thus, 


fa-!  f'a-Z  fan 

J o RJq  R 


or 


R 

R 1 R' 


E 


Since  Q =—  T = IT  is  the  number  of  coulombs  that  would 
R 

be  transferred  through  the  circuit  in  ^seconds  of  time  if  the 
current  had  its  full  value  of  I amperes  all  of  the  time  7,  the 

P 7 7 

formula  QT  = — T shows  that  during  the  rise  of  current  to 

R R 

E 

its  full  value  of  I = — amperes,  there  is  a deficit  of  coulombs 
R 


158 


ALTERNATING  CURRENTS 


transferred  through  the  circuit,  compared  with  an  equal  time 
of  full  current  flow;  and  that  this  deficit  has  a numerical 

value  of  Q'  = coulombs,  as  already  pointed  out.* 


The  total  energy  expended  in  the  circuit  during  the  time  of 
T seconds  while  the  current  is  rising  to  its  full  value  oi  I — 


amperes  is 


QtE  = EIT-LP 


R 


LI 2 . 


and,  of  this,  — — joules  goes  into  storage  in  the  magnetic  field. 


The  energy  expended  during  the  transient  period  of  the  rise  of 
current  is  therefore  less  than  the  energy  {LIT)  expended  in  the 
circuit  during  an  equal  length  of  time  with  the  current  at  its 
full  value.  This  is  analogous  to  the  energy  expended  on  a 
moving  body.  When  a given  force  is  externally  applied  to 
bring  the  body  in  T seconds  of  time  from  rest  to  full  speed  at 
which  the  force  is  just  balanced  by  frictional  opposition,  the 
distance  moved  by  the  body  and  the  total  work  (joules  or  foot- 
pounds) done  in  the  time  of  T seconds  are  less  than  would  have 
been  the  case  if  the  body  had  moved  at  its  maximum  velocity 
throughout  the  period  of  T seconds. 

If  the  external  source  of  energy  is  switched  out  of  the  circuit 
without  changing  the  resistance  and  self-inductance  when  the 
current  is  of  value  I amperes,  the  energy  equation  becomes 


0 = + L f^idi. 

Li 2 X-Z2  CT  • 

The  last  term  may  be  written — ; and— —is  equal  to  I i2Rdt , 

2 2 o 

which  shows  that  the  magnetic  field  (under  the  circumstances 
considered)  discharges  energy  equal  in  quantity  to  that  stored 
at  the  time  the  current  was  started,  and  that  this  energy  is  now 
all  converted  into  heat  during  the  flow  of  the  “extra  current.’’ 
The  energy  stored  in  the  magnetic  field  falls  off  as  the  current 
falls,  and  the  values  of  the  stored  energ}7  corresponding  to  the 
conditions  illustrated  in  Fig.  118  are  shown  by  the  curve  in 
Fig.  121.  The  rate  with  which  the  stored  work  is  given  out 
by  the  magnetic  field  at  each  instant  for  corresponding  condi- 
tions is  shown  by  the  curve  in  Fig.  121a. 


* Art.  46. 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  159 


The  voltage  formula  also  becomes 
0 = iR  -f-  L 


di 
Jt ’ 


Fig.  121.  — Energy  Stored  in  Magnetic  Linkages  at 
Each  Instant  during  Period  of  Falling  Current. 


or,  idt  = ^ di. 

R 

The  expression  idt  represents  the  coulombs  of  electricity  trans- 
ferred through  the  cir- 
cuit during  dt  seconds 
by  the  discharge  of  the 
energy  from  the  mag- 
netic field,  after  the  ex- 
ternal impressed  volt- 
age has  been  removed. 

That  is,  the  current 
does  not  fall  instantly 
to  zero  under  the  cir- 
cumstances when  the 
external  voltage  is 
switched  out  of  circuit 
without  changing  the  resistance  or  self-inductance,  but  the 
discharge  of  the  energy  from  the  magnetic  field  tends  to  main- 
tain it  briefly. 

The  total  coulombs 
conveyed  through  the 
circuit  by  this  “ extra 
current  ” may  be  ob- 
tained  by  integrating 
the  foregoing  formula 
from  the  instant  when 
the  external  voltage  is 
switched  out  of  the  cir- 
cuit (that  is,  from  zero 
Fig.  121  a. -Rate  °f  Change  of  Stored  Energy  at  of  time  and  j amperes 
Each  Instant  during  Period  of  Falling  Current.  r 

of  current)  through  the 
period  of  T seconds  during  which  the  current  falls  to  zero,  thus 


E = 0 VOLTS 
R = 5 OHMS 
[_  = 0.1  HENRY 


r«ft— #r 

RJi 


or. 


Q"  = 


R ■ 
LI 
R ' 


di, 


160 


ALTERNATING  CURRENTS 


It  will  be  observed  from  this  that  Q"  = Q',  which  is  to  say  that 
the  number  of  coulombs  conveyed  through  the  circuit  by  the 
“ extra  current  ” is  exactly  equal  to  the  deficit  in  the  coulombs 
conveyed  through  the  circuit  during  the  time  of  the  current 
rise,  as  previously  pointed  out,*  so  that  the  number  of  coulombs 
conveyed  through  the  circuit  as  a result  of  the  cycle  of  rise  and 
fall  of  current  does  not  differ  from  the  number  that  would  be 
conveyed  through  the  circuit  if  L were  zero  or  had  any  other 
value,  provided  R remained  unchanged  in  value.  The  time 
occupied  by  the  cycle  consisting  of  the  rise  of  the  current  to 


E 

its  full  fixed  value  of  — and  its  succeeding  fall  to  zero  upon 

switching  out  the  external  voltage  R is  affected  by  changing 
the  value  of  _L,  but  the  number  of  coulombs  transferred  through 
the  circuit  during  the  cycle  is  not  changed. 

The  gradual  decay  of  the  electric  current  under  the  condi- 
tions here  described,  accompanied  by  the  consumption  of  the 
energy  stored  in  the  magnetic  field,  is  analogous  to  the  decay  of 
the  velocity  of  the  moving  body  referred  to  above.  Upon  the 
removal  of  the  external  force  after  the  body  has  come  to  its  full 
speed,  the  body  would  continue  in  motion  with  gradually  fall- 
ing speed  and  finally  come  to  rest  when  the  energy  stored  in  its 
moving  mass  at  the  maximum  speed  had  all  been  consumed  in 
overcoming  the  frictional  resistance  and  been  converted  into 


heat. 

The  work  expended  in  a self-inductive  circuit  is  therefore 
manifested  by  (1)  the  iR  drop  relating  to  the  conversion  of 
energy  into  heat  and  (2)  the  counter- voltage  of  self-induction 
relating  to  the  storage  of  energy  in  the  magnetic  flux  by  a 
tendency  to  retard  a rising  current  and  accelerate  or  continue 
a falling  current.  The  effects  are  in  many  respects  analogous 
to  the  inertia  of  tangible  matter,  as  already  pointed  out,  in 

which  Mv,  and  are  respectively  the  energy,  the  mo- 

mentum, and  the  rate  of  change  of  momentum  (force)  acting 
in  the  mass  M when  moving  with  the  velocity  v ; while  in  the 
LP  . di 

electric  circuit,  , I/i,  and  L - — may  be  called  the  energy, 

2 dt 


momentum,  and  rate  of  change  of  momentum  (counter- voltage) 


* Art.  46. 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  1G1 


of  its  magnetic  field.  The  manifestations  of  the  electro-mag- 
netic inertia  are  exerted  through  the  linkages  of  the  magnetic 
lines  of  flux  with  the  conductors  of  the  electric  circuit. 

When  the  value  of  L varies  with  the  current,  as  when  iron 
is  magnetized  by  the  electric  current  under  consideration,  the 
energy  stored  in  the  magnetic  linkages  is  measured  by  the  value 
of  L corresponding  to  the  current  flowing  at  the  instant  under 


TT 

consideration;  and  the  value  of  L corresponding  to  1= — is 

Li 2 ^ 

the  correct  value  to  apply  in  the  expression  when  it  is 

desired  to  compute  the  energy  stored  during  the  rise  of 


U 

current  from  0 to  — amperes  in  a circuit  affected  by  a con- 


stant  impressed  voltage,  since  the  total  number  of  linkages 
only  is  required  to  obtain  the  stored  energy  and  not  the 
instantaneous  rates  at  which  they  are  created  or  disappear. 
The  phenomena  of  hysteresis  prevent  the  magnetic  field  from 
fully  discharging  its  energy  as  the  current  falls.  For  instance, 
if  the  current  is  caused  to  increase  from  Ix  to  I and  the  self- 
inductance changes  from  Ll  to  L , the  energy  that  is  delivered 
to  the  magnetic  field  by  virtue  of  the  increase  of  current  is 
Lp  l 1 2 

equal  to  — LJ-  • If  the  current  now  returns  to  the  value 


Ix  amperes,  hysteresis  may  prevent  the  magnetic  field  from 
assuming  its  first  value,  so  that  the  value  of  the  self-inductance 
has  the  different  value  of  Lv  In  this  case,  the  cycle  of  the 
rise  of  current  from  Ix  to  I and  its  restoration  to  Ix  has  caused 
an  increase  of  the  energy  stored  in  the  magnetic  field  which 

is  equal  to  ^2 — Lllh-.  If  the  current  changes  in  recurrent 

cycles  between  — I and  /,  the  conditions  in  the  magnetic  field 
also  become  cyclic,  and  a certain  amount  of  energy  is  converted 
into  heat  in  each  cycle  as  the  result  of  the  hysteresis  phenomena. 

If  a coil  is  wound  on  a closed  ring  of  soft  iron,  which  form 
exhibits  great  magnetic  retentiveness,  the  value  of  L is  very 
great  if  the  ring  is  magnetized  by  an  alternating  current. 
But  if  the  ring  is  magnetized  by  a rectified  periodic  cur- 
rent, that  is,  a periodic  current  which  is  unidirectional,  the 
apparent  value  of  L may  be  practically  the  same  as  though  the 
iron  core  were  not  present.  This  behavior  is  due  to  the  ring 


162 


ALTERNATING  CURRENTS 


continuously  retaining  the  magnetization  caused  by  the  maxi- 
mum current,  and  since  the  magnetism  in  the  core  therefore 
remains  constant  it  does  not  set  up  a counter-voltage.  By 
making  a transverse  cut  in  the  ring,  its  coercive  force  may  be 
reduced  so  much  that  the  effects  are  practically  the  same  for 
rectified  and  alternating  currents. 

Prob.  1.  A coil  with  a self-inductance  of  .02  of  a henry  has 
a steady  current  of  50  amperes  flowing  through  it.  What  is  the 
energy  of  the  magnetic  field? 

Prob.  2.  A magnetic  circuit  100  centimeters  long  and  200 
square  centimeters  in  cross  section,  the  material  of  which  has  a 
constant  permeability  of  1000  units,  is  excited  by  a coil  of 
500  turns  which  has  a resistance  of  10  ohms.  What  energ}^ 
has  the  magnetic  field  when  100  volts  are  steadily  impressed  at 
the  terminals  of  the  coil,  there  being  no  magnetic  leakage  ? 

Prob.  3.  A steady  current  of  two  amperes  flowing  in  a coil  of 
1000  turns  creates  a field  which  sets  up  1,000,000  lines  of  force 
through  each  turn  of  the  coil.  What  is  the  energy  of  the  field? 

Prob.  4.  A magnetic  circuit  is  excited  by  a coil  of  2000  turns 
in  which  a steady  current  of  20  amperes  flows  and  sets  up  a 
total  of  2,000,000  lines  of  force.  What  is  the  energy  of  the 
field,  if  all  the  lines  of  force  link  with  all  the  turns  of  the  coil? 

Prob.  5.  In  what  degree  does  changing  the  reluctance  of  a 
magnetic  circuit  affect  the  energy  stored  therein  by  an  exciting 
coil,  other  things  remaining  equal? 

Prob.  6.  The  energy  of  a certain  magnetic  field  is  20  joules. 
This  is  created  by  a magnetizing  coil  having  100  turns  carrying 
a steady  current  of  10  amperes.  What  is  the  self-inductance  of 
the  circuit  if  all  the  lines  of  force  of  the  field  link  with  all  the 
turns  of  the  coil? 

48.  The  Transient  Transfer  of  Electricity  in  Divided  Circuits. 
Application  to  a Shunted  Ballistic  Galvanometer.  — The  fact  that 
the  total  quantity  of  electricity  which  passes  through  a wire 
when  subjected  to  a transient  voltage  is  independent  of  the 
self-inductance  of  the  circuit  when  hysteresis  effects  are  absent, 
as  is  shown  above,  has  a bearing  upon  the  distribution  of  cur- 
rents in  divided  circuits.  With  no  external  disturbing  factors, 
it  is  apparent  that  where  a transient  voltage  is  impressed  upon 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  163 


parallel  circuits  of  different  inductances,  the  number  of  cou. 
lombs  which  flow  through  each  circuit  would  also  flow  were  the 
circuits  without  self-inductance,  but  the  phase  of  the  flow  in 
each  circuit  is  retarded  so  as  to  lag  behind  that  of  the  voltage 
by  an  amount  which  is  proportional  to  the  self-inductance  of 
the  circuit. 

This  reasoning  would  make  it  appear  that  shunting  a ballistic 
galvanometer  must  change  the  constant  in  the  ratio  of  the  re- 
sistances of  galvanometer  and  shunt  without  regard  to  their 
self-inductances,  as  is  true  when  steady  currents  are  in  ques- 
tion. This,  however,  is  not  correct,  because  one  of  the  factors 
in  this  problem  has  not  been  taken  into  account,  i.e.  the  move- 
ment of  the  needle  which  occurs  before  the  end  of  the  discharge 
generates  a counter-voltage  in  the  galvanometer  winding.  This 
reduces  the  proportion  of  the  discharge  which  passes  through 
the  galvanometer,  but  the  effect  is  due  to  the  needle  and  not  to 
the  relation  of  the  self-inductances  of  the  galvanometer  wind- 
ing and  the  shunt.  Assuming  that  the  number  of  lines  of  force 
due  to  the  needle  which  link  with  the  turns  of  the  winding  are 
proportional  to  the  sine  of  the  deflection  of  the  needle;  calling 
rg  and  L the  resistance  and  self-inductance  of  the  galvanometer 
winding ; rs  the  resistance  of  the  shunt  (the  inductance  of  the 
latter  being  assumed  negligible  on  account  of  its  being  wound 
with  doubled  wire) ; and  ig  and  is  being  the  respective  instan- 
taneous currents  : then  the  instantaneous  impressed  voltage  is 
e = isrs.  The  corresponding  instantaneous  value  of  the  IR  drop 
in  the  galvanometer  winding  is  igr0  and  this  is  equal  to  the 
impressed  voltage  less  the  corresponding  instantaneous  counter- 
voltages caused  by  self-induction  and  the  swing  of  the  needle. 

TI,eiefore:  . . (Ldi  kd(  siiw.)' 

\ dt  + dt 


where  & is  a constant  representing  the  summation  of  linkages 
of  magnetic  lines  of  force  from  the  needle  with  turns  of  the  gal- 
vanometer winding  when  the  needle  stands  perpendicular  to  the 
plane  of  the  winding.  Whence  isrsdt  — igrgdt  = Ldig+Jcd( sin  a) 

s*t  nt  /»o  /*a 

and  rs  I isdt—r„  ) igdt  = L I dig+ k I c?(sin «). 
do  d() 

This  is  rsqs  — rgqg  = k sin  «.  If  the  swing  of  the  needle  is  small, 


164 


ALTERNATING  CURRENTS 


then  sin  a is  sensibly  equal  to  2 sin^,  but  2 2Tsin-  = q T where 

2 2 

K is  the  ordinary  constant  of  the  ballistic  galvanometer.  Hence, 
Calling  Q the  total  discharge,  which  is  equal 
to  qs  + qg , this  becomes 


kqg 

7 srls  rg(lg 


% 


rs(Q  fdg)  rg1g  ’ and  qg 


Q'\s 


ra  + rs  + 


K 


This  discussion  shows  that  the  coulombs  of  a discharge  winch 
pass  through  a shunted  ballistic  galvanometer  are  less  than 
might  be  predicted  from  the  ratio  of  the  resistances;  that  is, 


do  ^ 


Qr, 


rg  + rs 


This  deficit  is  caused  solely  by  the  lines  of  force 


from  the  needle  cutting  the  galvanometer  coils  while  the  dis- 
charge is  passing,  and  its  value  is 

Qr, 


(?g  + rs) 


1 + j(rg  + 


The  shunted  ballistic  galvanometer  therefore  gives  readings 
which  are  too  small,  unless  the  duration  of  the  discharge  is 
very  small  compared  with  the  time  of  vibration  of  the  needle. 
But  it  must  also  be  remembered  that,  if  the  needle  is  removed 
from  the  galvanometer  or  clamped  in  a fixed  position,  the  divi- 
sion of  the  discharge  between  the  galvanometer  winding  and 
the  shunt  is  then  independent  of  the  self-inductance  of  the 
winding  and  shunt,  and  may  be  predicted  from  the  ratio  of 

resistances,  or  qa  = - • 

r 4—  x 
' g \ ' s 

Only  under  special  conditions  can  a single  coil  with  self- 
inductance be  substituted  for  the  coils  in  parallel  so  as  to 
produce  the  same  effect  as  the  latter  upon  transient  cui’rents 
of  every  duration.  These  conditions  are  fulfilled  when  the 

ratio  of  — is  constant  for  all  the  coils,  and  an  equivalent  coil 

R 

may  then  be  substituted  for  the  parallel  circuits.  In  this  case 
the  resistance  of  the  equivalent  coil  must  be 

- — = — -| -\ -(-  etc., 

R Rx  R2  i?3 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  1G5 


and  its  self-inductance  must  be 

i = T + A- 

L i,  L% 


H — — — (-  etc. 


These  values  make  — equal  to  the  constant  value  of  the  ratio 
K 

for  the  individual  coils.* 

49.  Effect  of  Eddy  Currents  and  of  Hysteresis  upon  the  Rise 
and  Fall  of  Current  in  a Self-inductive  Circuit.  — If  a circuit 
surrounds  or  lies  near  conducting  material,  the  rise  or  fall  of 
current  when  a voltage  is  applied  or  removed  is  modified  by 
the  magnetic  effect  of  eddy  currents  induced  in  such  conduc- 
tors.! The  eddy  currents  tend  to  quickly  rise  to  a maximum 
while  the  main  current  is  changing  most  rapidly,  and  then  to 
fall  gradually  as  the  main  current  approaches  its  full  value. 
Eddy  currents  are  similar  to  the  secondary  currents  of  a trans- 
former | and  are  induced  in  the  same  way.  As  in  the  trans- 
former, an  additional  current  just  sufficient  to  neutralize  the 
magnetic  effect  of  the  secondary  currents  at  the  instant  flows 
in  the  main  or  primary  circuit.  This  extra  neutralizing  ele- 
ment of  the  main  current  causes  the  curve  of  current  to  rise 
more  rapidly  when  the  voltage  is  applied  and  fall  more  quickly 
when  the  voltage  is  withdrawn ; that  is,  energy  is  expended 
not  only  in  building  up  the  magnetic  field  and  in  i2R  loss  in 
the  main  circuit,  but  also  in  the  eddy  current  circuits,  as  the 
main  current  rises,  and  while  the  main  current  is  falling,  the 
energy  yielded  from  the  magnetic  field  is  absorbed  by  i'2R  losses 
in  the  main  circuit  and  also  in  the  eddy  current  circuits.  This 
transfer  of  energy  from  the  main  circuit  to  the  eddy  current 
circuits  has  the  same  effect  on  the  rise  and  fall  of  the  current  in 
the  main  circuit  as  would  be  produced  by  increasing  to  an  equal 
degree  the  energy  changed  into  heat  by  the  i2R  loss  in  the 
main  circuit  and  excluding  eddy  current  effects.  Any  trans- 
fer of  energy  by  induction  from  the  main  circuit  while  the 
main  current  is  changing  has  the  same  effect  of  hastening  the 
change  of  the  main  current,  and  thus  reducing  the  apparent 
effect  of  the  self-induction.  It  is  therefore  possible  to  surround 
a self-inductive  circuit  with  conducting  material  in  such  a 
manner  as  to  largely  mask  the  presence  of  self-inductance. 

* The  subject  of  circuits  in  parallel  is  fully  treated  in  later  pages, 
t Art.  111.  J Art.  23  and  Chap.  X. 


166 


ALTERNATING  CURRENTS 


On  account  of  the  counter-magnetizing  effect  of  eddy  cur- 
rents the  lines  of  force  set  up  by  the  main  current  tend  to 
crowd  outside  of  the  eddy  current  circuits,  which  decreases  the 
average  cross  section  of  their  path  and  hence  increases  the 
reluctance.  This  effect  of  eddy  currents  in  apparently  screen- 
ing the  interior  of  an  iron  core  from  magnetic  influences  de- 
creases the  number  of  lines  of  force  per  ampere  in  the  circuit 
and  consequently  also  decreases  the  apparent  self-inductance. 
It  consequently,  also,  causes  an  acceleration  in  the  rise  or  fall 
of  the  current.  If  the  circuit  surrounds  an  iron  core,  this 
effect  may  be  quite  large  unless  the  core  is  finely  laminated. 

The  phenomena  of  hysteresis  and  residual  magnetism  tend 
to  produce  a smaller  average  rate  of  change  of  the  magnetic 
field  set  up  in  an  iron  core  when  the  exciting  current  is  with- 
drawn than  occurs  when  the  exciting  current  is  introduced  in 
the  circuit  ; and  a current  therefore  dies  away  more  quickly 
than  it  builds  up  in  a circuit  with  an  iron  core,  even  though 
the  source  of  voltage  is  switched  into  the  circuit  and  later 
removed  without  altering  the  resistance  of  the  circuit.  The 
change  of  permeability  of  iron  for  different  magnetizations, 
which  results  in  a change  in  number  of  magnetic  lines  set  up 
per  ampere  in  the  electric  circuit,  causes  a distortion  of  the 
curves  of  rising  or  falling  current,  as  already  pointed  out  in 
the  latter  part  of  Art.  46.  This  effect  becomes  particularly 
noticeable  when  the  magnetic  circuit  is  completed  through 
iron,  as  in  the  case  of  a closed  ring  core  or  the  transformer 
core  illustrated  in  Fig.  45,  as  any  air  gap  in  the  path  of  the 
magnetic  lines  of  force  introduces  a space  of  relatively  large 
reluctance  which  is  independent  of  the  current. 

50.  High  Voltage  generated  on  Breaking  a Self-inductive 
Circuit.  — The  condition  under  which  the  curves  of  rising  and 
falling  current  are  logarithmic  and  of  exactly  the  same  dimen- 
sions when  voltage  is  applied  to  and  withdrawn  from  a circuit 
requires  that  the  resistance  as  well  as  the  self-inductance  of  the 
circuit  shall  remain  constant.  If  the  circuit  is  quickly  broken, 
by  opening  a switch  or  otherwise,  it  is  a well-known  fact  that 
the  counter-voltage  rises  much  higher  than  the  original  impressed 
voltage , frequently  rising  to  many  times  its  value.  The  extreme 
severity  of  the  shock  which  may  be  received  upon  breaking  a 
circuit  of  large  inductance  attests  the  fact.  This  is  due  to 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  167 


the  exceedingly  large  increase  of  resistance  in  the  circuit  which 
is  introduced  by  the  break.  This  increase  in  resistance  causes 
the  current  to  fall  off  more  quickly,  and  hence  produces  a 
greater  rate  of  change  of  magnetism.  However,  as  before,  the 
energy  given  up  by  the  field  must  be 


where  q is  the  current  after  the  break,  since  the  same  number 
of  magnetic  linkages  disappear  from  the  circuit,  and  the  energy 
given  up  must  therefore  be  equal  to  the  energy  stored  in  the 
circuit  when  the  full  current  I was  flowing ; and,  also 


where  is  the  new  resistance  of  the  circuit,  including  the 
break,  assumed  to  be  constant,  and  cp  is  the  coulombs  of  the 
discharge.  The  current  is 


for  small  values  of  t.  The  total  number  of  coulombs  trans- 
ferred is,  therefore,  less,  and  the  induced  voltage  greater,  upon 
breaking  a circuit  than  upon  making  it.*  If  hysteresis  is  present, 
the  field  may  not  be  totally  discharged,  and  the  amount  of 
energy  and  quantity  of  electricity  will  then  not  be  so  great 
as  indicated  by  the  formula,  as  is  explained  in  the  previous 
article. 

Prob.  1.  A certain  electric  circuit  supports  a magnetic  field 
which  is  charged  with  an  energy  of  5 joules.  This  circuit  is 
opened  by  introducing  resistance  increasing  to  infinity  in  .01 
of  a second.  What  is  the  average  power  exerted  by  the  mag- 
netic field  during  the  time  of  the  break,  assuming  that  the 
magnetism  falls  to  zero  ? Give  the  answer  to  this  and  the 
following  problems  in  watts  and  neglect  the  effect  of  hysteresis 
and  eddy  currents. 

* Formulas  for  further  expressing  the  conditions  when  the  resistance  of  a 
circuit  is  changed  are  given  in  Art.  55. 


q = Ie  i < 7e  z, 

for  finite  values  of  t ; and  the  induced  voltage  is 


168 


ALTERNATING  CURRENTS 


Prob.  2.  If  6,000,000  lines  of  force  linking  the  turns  of  a 
coil  of  1000  turns  are  set  up  by  a steady  current  of  5 amperes 
in  the  coil,  what  is  the  average  power  exerted  by  the  discharge 
of  the  magnetic  field  if  the  exciting  circuit  is  broken  in  .01 
second  by  introducing  resistance  up  to  infinity,  and  the  mag- 
netism falls  to  zero  ? 

Prob.  3.  A magnetic  circuit  having  an  average  permeability 
of  1000  units,  a length  of  200  centimeters,  and  a cross  section 
of  50  square  centimeters  is  excited  without  magnetic  leakage 
by  a coil  of  1000  turns  having  a resistance  of  25  ohms.  A 
steady  voltage  of  1000  volts  is  impressed  at  the  terminals  of 
the  coil.  What  is  the  average  power  exerted  by  the  discharge 
of  the  magnetic  field  when  the  exciting  circuit  is  broken 
through  an  arc  maintained  .01  of  a second? 

51.  Capacity.  — All  insulated  conductors  have  the  property 
of  being  able  to  hold  a charge  of  electricity.  When  an  insu- 
lated conductor  is  connected  to  a source  of  a different  potential, 
electricity  will  flow  to  it  or  from  it,  until  its  potential  is  the 
same  as  that  of  the  source.  The  measure  of  the  charge  or 
quantity  of  electricity  which  is  held  by  the  conductor  when  at 
unit  potential  is  its  electrostatic  Capacity,  and  the  unit  of  ca- 
pacity may  be  defined  as  the  capacity  of  a conductor  which  con- 
tains a unit  charge  of  electricity  when  at  unit  potential.  The 
practical  unit  of  capacity  is  the  capacity  of  an  isolated  conductor 
which  ivould  contain  a charge  of  one  coulomb  when  at  a potential 

of  one  volt.  This  is  called  a Farad,  after  Faraday.  It  is  — jj9 

times  as  large  as  the  unit  of  capacity  of  the  C.  G.  S.  set  of 
units.  The  farad  is  too  large  a unit  of  capacity  to  be  conven- 
ient in  practice,  and  the  Microfarad,  or  millionth  of  a farad,  is 
commonly  used  as  the  unit  of  measurement.  The  capacity  of 
a conductor  depends  upon  its  conformation  and  surroundings. 

Two  adjacent  conducting  bodies  brought  to  different  poten- 
tials and  isolated  from  external  influences  assume  equal  and 
opposite  charges.  In  this  case,  the  capacity  is  measured  in 
terms  of  the  difference  of  potential  between  the  conductors  and 
the  charge  on  each.  The  capacity  is  one  farad  when  a differ- 
ence of  potential  between  the  conductors  of  one  volt  produces 
a charge  of  one  coulomb  on  each,  — the  charges  being  (relative 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  169 

to  each  other)  positive  on  the  conductor  of  higher  potential  and 
negative  on  the  conductor  of  lower  potential. 

Capacity  effects  caused  by  differences  of  potential  between 
neighboring  conductors  or  parts  of  the  same  conductor,  are  of 
constant  occurrence  and  much  importance  in  electrical  engineer- 
ing structures.  In  most  of  such  instances  the  influences  are 
complex  because  many  conductors  of  different  potentials  and  of 
different  shapes  and  space  relations  relative  to  each  other  may 
be  involved ; but  most  of  these  instances  may  be  reduced  to 
approximate  analytical  processes  by  considering  the  conductors 
pair  by  pair  and  thus  learning  their  influence  on  each  other. 

The  term  Condenser  is  applied  to  any  pair  of  insulated  con- 
ductors having  an  appreciable  capacity,  although  it  is  more 
strictly  used  to  designate  a combination  of  sheets  of  conducting 
material,  insulated,  and  laid  together  with  the  alternate  layers 
connected  in  parallel.  In  the  following  discussion  the  term 
condenser  will  be  used  in  its  broader  sense. 

The  Dielectric  of  a condenser  is  the  insulating  medium  which 
surrounds  the  conductors  ; and  its  dimensions  and  quality  deter- 
mine the  condenser  capacity.  The  capacity  is  directly  propor- 
tional to  the  area  and  inversely  as  the  thickness  of  the  dielectric 
and  depends  upon  the  Specific  inductive  capacity,  or  Dielectric 
constant  of  the  dielectric  material.  The  Charge  of  a condenser 
is  the  quantity  of  electricity  comprised  on  either  conductor  or 
electrode  of  the  condenser. 

It  should  be  kept  clearly  in  mind  that  the  capacity  of  a con- 
ductor or  circuit  is  a quality  of  the  circuit  and  is  not  depend- 
ent upon  the  current  flowing  or  the  voltage  applied.  In  the 
same  way  the  capacity  of  a vessel  to  hold  a liquid  is  a quality 
of  the  vessel,  dependent  upon  the  conformation  of  the  vessel, 
and  irrespective  of  the  amount  of  liquid  that  may  be  in  the 
vessel  or  the  rate  at  which  it  is  running  in  or  out. 

From  the  foregoing  definitions  of  capacity  is  at  once  derived 
the  fundamental  relation, 

Q = CE, 

where  Q represents  the  quantity  of  electricity  (that  is,  the 
coulombs)  in  the  charge  of  a condenser,  C the  capacity  in 
farads,  and  E the  voltage  between  the  conductors  or  electrodes 
of  the  condenser. 


170 


ALTERNATING  CURRENTS 


Prob.  1.  A condenser  is  charged  by  bringing  its  electrodes 
to  a difference  of  potential  of  100  volts.  What  is  its  capacity 
if,  under  that  voltage,  its  charge  is  .002  coulomb? 


52.  Conditions  of  Establishment  and  Termination  of  Current  in 
a Circuit  containing  Capacity  and  Resistance  in  Series.  — a.  Cur- 
rent and  Rise  of  Charge  upon,  Impressing  Constant  Voltage.  — The 
work  done  during  time  dt  on  a circuit  containing  resistance 
and  capacity  at  instant  t after  a constant  voltage  E is  im- 
pressed thereon  is 

Edq  = Eidt  = edq  + Ri2dt , 


where  e,  i,  and  q are  the  instantaneous  values  at  instant  t of  the 
voltage  of  the  charge,  current  flowing  into  the  condenser,  and 
quantity  of  the  charge.  The  last  member  contains  two  terms, 
of  which  the  first  is  the  work  stored  in  the  increment  dq  coulombs 
of  the  charge  which  is  put  in  the  condenser  during  the  time  dt 
seconds,  and  the  second  is  the  woi’k  converted  into  heat  by  dq 
coulombs  flowing  through  the  resistance  R of  the  circuit.  If 
this  equation  is  divided  by  idt  = dq,  there  results  an  equation 

E = e -(-  Ri , 


of  voltage 


or 


From  these  equations  the  charge  at  any  instant  may  be 
determined  when  the  applied  voltage  E is  constant  during  the 
process  of  charging.  The  equation  may  be  put  in  the  form 

dq  dt 

q-CE  = “ EC1 


whence 

Integrating  gives 


log  (q-  CE ) 


V dt 
o RC' 


and  q — CE  = — CEe  RC. 

Therefore  since  CE  = Q,  where  Q is  the  final  value  of  the  charge, 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  171 


q = Q(l  — e RC ), 


and 


i_dq_Q  JrC' 
dt  RC 


Figure  122  shows  the  logarithmic  curves  of  the  rise  of  charge 
in  a condenser  in  a particular  circuit  and  the  current  flowing 


Q t_ 

v e so  during  the  same  process.  It  is 


R2C2 


into  the  condenser  (that  is,  the  rate  of  change  of  the  charge) 
while  the  charge  is  reaching  its  full  value  Q after  100  volts  are 
impressed  on  the  circuit.  Figure  122  a shows  the  rate  of  change 

of  the  current  — = 
dt 

to  be  noted  that 
the  current  at  the 
instant  of  intro- 
ducing the  voltage 
E into  the  circuit 
(i.e.  t = 0)  is  i = 

Q E 

RC=R  ampereS’ 
and  it  is  therefore 
instantaneously 
equal  to  the  cur- 
rent that  the  vol- 
tage E would  cause 
to  flow  through  the  resistance  R of  the  circuit  if  the  condenser 
were  absent. 

b.  Current  and  Fall  of  Charge  on  Withdrawal  of  Impressed 
Voltage.  — During  the  discharge,  assuming  that  the  impressed 


Fig. 


122a.— Curve  showing  Rate  of  Change  of  Current 
during  the  Period  of  Charging. 


172 


ALTERNATING  CURRENTS 


voltage  is  withdrawn  from  the  circuit  without  changing  R or 
(7,  the  impressed  voltage  E is  zero,  and  therefore 

,dq 


0 = — - 

c 


R 


dt ’ 


dq 

dt 

or 

9 ~ 

= ~TlC ’ 

and 

4 — > 

II 

so  1^ 
1 

II 

Integrating  gives 

log  q f = 

7- 

RCjq 

or 

log«“ 

t 

RO ’ 

whence 

9 = 

Qe~*a, 

and 

Q-  €~ 

dt 

RC 

Figure  123  shows  the  logarithmic  curves  of  falling  charge  after 
the  impressed  voltage  has  been  removed  and  of  current  flowing 

during  the  period  of  dis- 
charging. Figure  123  a 
shows  the  rate  of  change 

of  current,  — , during1 
dt  ° 

the  same  period. 

From  the  equations 
showing  the  charging 
and  discharging  cur- 
rents, which  are  of 
forms  similar  to  those 
showing  the  rise  and 
fall  of  currents  in  cir- 
cuits containing  resist- 
ance and  self-induct- 
ance, it  is  seen  that 
the  curves  of  charge 
and  discharge  in  cir- 
cuits containing  resist- 


-.10 


- 20 


6 QBMS 
60  MlCROftj 


Fig.  123.  — Curves  of  Discharge  of  a Condenser. 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  173 


Fig.  123a.  — Curve  showing  Rate  of  Change  of  Cur- 
rent during  the  Period  of  Discharging. 


ance  and  capacity  in  series  are  logarithmic  when  an  unvarying 
voltage  is  impressed  on  the  system.  (See  Figs.  122  and  123.) 
In  many  cases  of  practice  RC  is  so  small  that  the  charge  and 
discharge  of  a condenser  are  practically  instantaneous,  but  in 
other  cases  the  dura- 
tion of  the  rise  or  fall 
of  the  charge  may  be 
quite  appreciable. 

Some  condensers  have 
a quality  which  has 
some  analogy  to  mag- 
netic hysteresis  and 
which  is  often  termed 
dielectric  hysteresis 
from  the  fact  of  its 
being  due  to  a charac- 
teristic of  the  dielectric.  When  hysteresis  is  present,  the  con- 
denser does  not  at  once  proceed  to  a complete  discharge  when 
the  impressed  voltage  is  removed  from  its  terminals,  but  it 
retains  a residual  charge.  The  amount  of  energy  retained  in 
this  residual  charge  is  a measure  of  the  amount  of  energy 
absorbed  by  the  condenser  on  account  of  hysteresis,  and  this 
energy  may  be  converted  in  each  cycle  into  heat  if  the  con- 
denser is  subjected  to  alternating  charges  and  discharges.  It 
is  probable  that  this  effect  is  due  to  a small  part  of  the  electric 
charge  soaking  into  the  dielectric  (especially  into  those  layers 
near  the  condenser  electrodes)  and  then  soaking  out  again, 
because  of  the  imperfect  insulating  qualities  of  the  dielectric, 
rather  than  to  molecular  phenomena  strictly  analogous  to  mag- 
netic hysteresis  ; and  the  heating  of  the  dielectric  which  is 
sometimes  observed  as  the  result  of  repeated  charging  and  dis- 
charging may  be  ascribed  to  the  I2R  loss  due  to  the  current 
flow  in  the  dielectric  which  results  from  the  aforesaid  soaking 
in  and  out  of  the  charge.  The  effect  is  not  ordinarily  very  im- 
portant in  engineering  practice. 


Prob.  1.  A condenser  of  50  microfarads  capacity  and  negli- 
gible internal  resistance  is  connected  in  series  with  a wire 
having  a resistance  of  10  ohms.  Draw  the  curve  of  charge  in 
this  circuit  when  100  volts  are  impressed ; and  of  discharge 


174 


ALTERNATING  CURRENTS 


when  the  impressed  voltage  is  removed  without  breaking  the 
circuit. 

Prob.  2.  What  quantity  of  charge  has  the  condenser  of 
problem  1 received  at  the  instant  when  the  impressed  voltage 
has  been  applied  for  R C seconds? 

Prob.  3.  What  quantity  of  charge  remains  in  the  condenser 
of  problem  1,  RC  seconds  after  the  removal  of  the  impressed 
voltage? 


c.  Current  and  Charge  when  the  Impressed  Voltage  is  a Sine 
Function  of  the  Time.  — Under  these  conditions  the  voltage 
equation  of  the  circuit  becomes 


e=f(f)  = em  sin  cot  = Ri  + & 

0 


=Ri+fiM 


c 

This  is  a linear  differential  equation  of  the  first  order,  the  solu- 
tion of  which  is  * 


i =- 


\ R2  -I 1 

' ' 4 7T2f  2 C2 


sin  («  — 0)  + Bxe  nc, 


where  a = c of,  oo  = 2 7 rf  and  6 — — tan-1(  — - R )• 

\coC 

In  a similar  manner  the  value  of  q is  found  to  be 


r =J  idt  = — 


— 

' 4 t rf2C‘ 


cos  («  — d)  + B2e  ac- 


After  a short  time  the  exponential  term  of  the  equations  be- 
comes inappreciable  and  may  be  neglected.  The  effect  of  this 
exponential  term  will  be  discussed  later,  f The  current  then 
becomes  a sine  function  of  the  time  and  leads  the  impressed 


voltage  by  an  angle  0 whose  tangent  is 


1 

2 irfC 


h-  R , instead  of 


* Murray’s  Differential  Equations,  p.  26. 
t Art.  69. 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  175 


lagging  behind  it,  as  in  the  case  of  a circuit  containing  induct- 
ance and  resistance.  The  current  has  a maximum  value 


= 


or  an  effective  value  of 


^2+47 r2f2C2 


/= 


E 


+ 4 7T2f2C2 


in  which  E is  the  effective  value  of  the  sinusoidal  impressed 
voltage.  The  denominator  of  this  expression  is,  as  in  the  case 
of  the  inductive  circuit,  called  the  Impedance  of  the  circuit. 
This  impedance  is  composed  of  the  square  root  of  the  sum  of 
two  terms,  one  of  which  is  the  square  of  the  electrical  resistance 


and  the  other  the  square  of  the  expression 


This  latter 


2 TrfC 

term  is  called  the  Reactance  of  the  circuit,  and  corresponds  in 
this  case  to  the  term  similarly  designated  in  the  circuit  contain- 
ing inductance  and  resistance. 

53.  The  Energy  of  a Charged  Condenser.  — As  a condenser  is 
charged,  a certain  amount  of  work  is  done  in  raising  the 
potential  of  the  charge.  During  the  time  dt  this  is  equal  to 


dw  = (eidt  = edq ) : Cede , 


in  which  e is  the  voltage  of  the  charge,  i the  current  flowing 
into  the  condenser,  and  q the  charge  in  the  condenser  at  the 
instant  t ; from  which,  by  integration. 


w __CE2  EQ 

2 2 ’ 


where  E and  Q are  respectively  the  final  values  of  the  voltage 
and  the  quantity  composing  the  charge. 

This  represents  a certain  amount  of  work  which  is  stored  in 
the  condenser  when  its  charge  is  increased  from  zero  to  Q 
coulombs.  When  the  condenser  is  discharged,  an  equal  amount 

CE2 

of  work  is  returned  to  the  circuit.  The  expression  — — is 

similar  to  that  giving  the  work  stored  in  a compressed  spring, 
NF2 

W = --  -,  where  N is  the  compressibility  of  the  spring  meas- 


176 


ALTERNATING  CURRENTS 


ured  by  the  distance  compressed  per  unit  of  force,  and  F is 
the  force  applied  to  perform  the  compression.  These  expres- 
sions are  proportional  to  the  square  of  impressed  electrical  or 
physical  pressure  and  the  stored  energy  is  truly  potential. 

LI 2 

These  equations  thus  differ  from  the  expressions  — — - and 
— —4,  which  are  dependent  upon  momentum  instead  of  pressure 
and  represent  kinetic  energy. 

As  we  know  the  current  and  the  value  of  the  charge  at 
each  instant  during  the  rise  of  the  charge,  it  is  practicable  to 

d 2 

compute  the  value  of  the  energy,  w — Ce2  = | , stored  in 

v 

the  condenser  at  each  instant  during  the  rise  of  charge,  and 
the  rate  at  which  the  energy  is  changing  at  each  instant. 
These  values  are  exhibited  in  the  curves  of  Figs.  124  and  124  a 
for  the  conditions  of  the  circuit  corresponding  to  Fig.  122. 


Fig.  124. — Energy  stored  in  Condenser  during  Period  of  Charging. 


The  energy  stored  in  the  condenser  at  each  instant  during 
the  fall  of  the  charge,  and  the  rate  at  which  the  condenser  gives 
out  energy  (the  rate  of  change  of  the  stored  energ}  ) during  the 
fall  of  the  charge,  for  circuit  conditions  corresponding  to  those 
indicated  in  Figs.  123  and  123a  are  shown  in  Figs.  125  and  125a. 

54.  Vector  Diagrams  showing  the  Voltage  and  Current  Rela- 
tions in  a Circuit  containing  R and  C-  — Expression  in  Form  of 
Complex  Quantities.  — When  a condenser  is  connected  to  a 
source  of  alternating  voltage,  as  indicated  in  Fig.  126,  a cur- 
rent will  flow  into  and  out  of  the  condenser.  The  value  of  the 
current  at  any  instant  is  proportional  to  the  rate  of  change  of 
the  voltage  impressed  on  the  capacity,  because  the  charge  in 
the  condenser  at  any  instant  is  proportional  to  the  charging 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  177 


E = 1 00  VOLTS 
R = 5 OHMS 

C=  50  MICROF. 


.0002 


.0004 


.0006 


.0008 


voltage  then  acting  on  the  capacity,  and  the  rate  at  which  the 
charge  changes  must  be  proportional  to  the  rate  at  which  the 
voltage  changes.  The  rate 
of  change  of  the  charge  is 
equal  to  the  number  of 
coulombs  flowing  per  se- 
cond into  or  out  of  the 
condenser,  and  is  there- 
fore equal  to  the  current  o 
flowing  into  or  out  of  the  Fig.  124  a. — Rate  of  Change  of  Energy  stored  in 
, ' mi  i Condenser  during  Period  of  Charging. 

condenser.  1 he  condenser 

de 

current  (ic)  at  any  instant  is  represented  by  ic  = c'~,  and  since, 

when  the  alternating  voltage  is  sinusoidal, 
e — em  sin  a = em  sin  cot, 

where  em  is  the  maximum  voltage  acting  upon  the  condenser 
during  a cycle,  co  is  the  angular  velocity  or  advance  of  the 
cycle,  and  t the  time  measured  from  the  beginning  of  a cycle, 

clc 

— = emco  cos  cot. 
dt 

But  co  = 2 rrf, 

where  /is  the  frequency  of  the  cycles. 


Therefore, 


and 


= em  cos  a -4- 


1 


TTfC 


= 2 7j -fCem  cos  a 


em  cos  a = i.  x 


2 7 rfO 

The  voltage  ec  of  the  condenser  charge,  which  is,  of  course, 
equal  and  opposite  to  the  voltage  e impressed  on  the  plates, 

may  be  called  the 
Capacity  voltage  or 
Condenser  voltage. 
This  voltage  is 
purely  reactive,  due 
to  the  charge  in  the 
condenser.  That  it 
must  be  90°  in  ad- 
vance of  the  current 
when  the  impressed 


E — o 

R = 5 OHMS 
C = 50  MICROF. 


.0002 


.0008 


Fig.  125.- 


.0004  1 .0000 

SECONDS  ’ 

-Energy  stored  in  Condenser  at  Each  Instant  Voltage  is  SlllUSOl  a 
during  Period  of  Discharging.  may  be  readily  seen 


178 


ALTERNATING  CURRENTS 


SECONDS 

.0004  .0006 


.0008 


from  the  formula  above  and  the  reactions  that  occur  in  the  cir- 
cuit. When  a sinusoidal  voltage  impressed  at  the  terminals  of  a 

resistanceless  condens- 
er is  rising,  a current 
flows  into  the  condenser. 
This  current  is  a posi- 
tive maximum  at  the 
instant  the  impressed 
voltage  passes  through 
zero  in  changing  from 
negative  to  positive 
values,  for  the  rate  of 
Fig.  125  a.  — Rate  of  Change  of  Energy  stored  in  change  of  voltage  is 
Condenser  during  Period  of  Discharging.  then  a positive  maxi- 

mum. When  the  impressed  voltage  passes  through  its  maxi- 
mum point,  its  rate  of  change  is  zero,  and  the  current  at  that 
instant  is  therefore  zero.  When  the  voltage  is  falling,  a cur- 
rent flows  out  of  the  condenser ; that  is,  in  the  direction  which 
discharges  the  condenser.  There-  condenser 

fore,  the  current  flowing  is  90°  in 
advance  of  the  voltage  impressed  on 
the  capacity.  The  capacity  voltage 
is  a counter-voltage  caused  by  the 
potential  of  the  charge,  and  is  equal 
and  opposite  to  the  voltage  impressed 
on  the  capacity,  and  is  90°  in  advance  of  the  current. 

From  the  formula  for  instantaneous  current  in  the  condenser 
the  maximum  current  is  seen  to  be 


ALTERNATOR 


Fig.  12(5. — Diagram  showing  an 
Alternator  connected  to  a Ca- 
pacity Circuit. 


im  = 2 7rfCem, 
and  the  effective  current, 

Ic  = 2 irfCE, 

where  E is  the  effective  value  of  the  voltage  impressed  on 
the  capacity.  This  current.  Ic  being  90°  ahead  of  the  phase  of 
the  voltage  impressed  on  the  capacity  (which  is  180°  behind  the 
capacity  voltage),  the  phase  of  the  active  voltage  which  drives 
the  current  through  the  resistance  of  the  circuit  is  also  90° 
ahead  of  the  voltage  impressed  on  the  capacity.  The  voltage 
impressed  upon  the  series  circuit  containing  capacity  and  re- 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  179 


sistance  must  therefore  comprise  two  components,  one  equal 
to  the  IR  drop  in  the  circuit  and  the  other  equal  and  opposite 
to  the  capacity  vol- 
tage. A phase  dia- 
gram of  these,  ac- 
companied by  the  D 
corresponding 
sinusoidal  curves, 
is  shown  in  Fig. 

127,  where  00, 

CA.  and  OA  are  ®'IG-  127.  — Phase  Diagram  of  Voltages  in  a Circuit  con- 
. taming  Resistance  and  Capacity  in  Series, 

respectively  the 

IR  drop,  the  component  equal  but  opposite  to  the  capacity 
voltage,  and  the  voltage  impressed  on  the  circuit.  The  triangle 
of  voltages  Er,  Ex,  E,  as  shown  in  Fig.  128,  is  derived  from  this 
diagram,  and  it  may  be  observed  that 

E=-JE?  + E?-, 


and  also  that  the  angle  of  lead,  i.e.  the  angle  by  which  the 
current  and  IR  drop  lead  the  voltage  impressed  on  the  circuit, 
is  the  angle  0 , with 


tan  0 — 


--ffr 

Er 


: = E 


The  angle  0 is  here  called  negative  because  it  relates  to  a 

leading  current,  and  the 
corresponding  angle  for  a 
lagging  current  has  al- 
ready been  called  positive. 

The  triangle  of  voltage 
(Fig.  128),  taken  from  the 
phase  diagram  in  Fig.  127, 
is  similar  to  the  triangle 
for  a self-inductive  circuit 
illustrated  in  Fig.  119,  except  that  the  current  in  the  circuit 
and  IR  drop  lead  the  impressed  voltage  instead  of  lagging 
behind. 

It  must  be  carefully  borne  in  mind  that  in  self-inductive  and 
capacitjr  circuits  the  self-inductive  voltage  and  the  capacity 
voltage  act  under  the  same  laws  as  any  other  electric  voltages. 
The  impressed  voltage , therefore,  must  comprise  ttvo  rectangular 


Fig.  128. — Triangle  of  Voltages  in  a Circuit 
containing  Resistance  and  Capacity  in  Series. 


180 


ALTERNATING  CURRENTS 


components , one  equal  and  opposite  to  the  reactive , and  the  other 
equal  to  the  active  voltage  ( i.e . the  IR  drop).  The  triangles 
of  Figs.  119  and  128  indicate  this.  The  vertical  sides  Ex 
of  the  triangles  are  vector  components  of  the  hypothenuses 
which  represent  the  voltages  impressed  on  the  circuits,  and  are 
respectively  equal  and  opposite  to  the  self-inductive  or  capacity 
voltages  in  the  circuit,  while  the  active  voltages  (J72  drop) 
represented  by  the  horizontal  sides  of  the  triangles  complete 
the  respective  right-angled  vector  triangles. 

The  vector  representing  the  impressed  voltage  in  a circuit 
containing  resistance  and  capacity  may  be  indicated  by  the 
complex  quantity 

E=Er-jEx, 

E—E  (cos  6 — j sin  O'). 


E is  the  scalar  value  of  E.  The  scalar  value  of  E and  the 
angle  6 are  determined  in  the  same  manner  as  explained  in 
Art.  40,  as  is  also  the  sum  of  several  voltages  in  series. 


Since 


6 = tan  1 


the  angle  of  lag  6 is  negative,  which  follows  from  the  fact  that 
the  current  leads  the  impressed  voltage. 

The  expression 

E = E (cos  0 —j  sin  6) 
obviously  may  be  written 


E = E (cos  ( — 6)  +j  sin  ( — 0)). 


55.  The  Effect  of  Introducing  Resistance  in  a Continuous  Cur- 
rent Circuit ; Opening  the  Circuit.  — It  has  already  been  pointed 
out  in  Art.  50  that  the  voltage  must  rise  at  the  breaking  of  a 
direct  current  circuit  possessing  self-inductance.  Breaking 
such  a circuit  has  the  effect  of  very  rapidly  introducing  resist- 
ance until  the  resistance  becomes  infinite,  with  a correspond- 
ingly  rapid  forced  decrease  of  the  current.  In  case  of  a closed 
circuit  of  resistance  R and  self-inductance  L upon  which  a 
steady  voltage  E is  impressed,  the  equation  representing  the 
conditions,  while  the  circuit  is  in  process  of  being  broken,  is 

J!=z|+[i8+/(0]j, 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  181 


in  which  /(£)  is  a function  of  time  measured  from  the  instant 
of  starting  the  process  and  represents  the  rate  of  introducing 
extra  resistance  to  open  the  circuit. 

This  may  be  transformed  into 


' di  [ig  +/(Q]  i _ E 
dt  L L 


The  rate  at  which  extra  resistance  is  to  be  introduced  at 
breaking  the  circuit — namely,  the  form  of  f(t)  — must  be  known 
before  this  equation  can  be  solved.  Assuming  that  the  resist- 
ance is  added  as  a simple  function  of  time,  then  f(f)  = at , in 
which  a is  a constant,  and  the  equation  becomes 


di  (R  + at  ) i _ E 

It  L ~ L 

. / — - ■ di  du  di 

Tutting  R + at  — — 2 aL  and  — 


(2) 

Va  di 


this  becomes 


di 


— 2 ui=  — 

du  V — 2 aL 


dt  dt  du  V — 2 L du 
2 E 


(3) 


and  the  solution  is  of  the  form* 

2 E 


— J*2  udu 

ie 


= --±2=  C £~Pudu  du  + K. 

I-taLd 


Hence,  i = 


2 E 

— t 

V — 2 aL 


I*1 

"X 


e u2du  = — ■ 


2 E 


V — 2 aL 


Ce~*du-  Ce~u2du\.  (4) 
To  Ju  J 


When  this  is  integrated  by  parts  and  approximated  by  neglect- 
ing evanescent  terms  and  terms  containing  the  reciprocal  of 
(u?)n,  the  equation  reduces  to 

E 


i = 


R -|-  at 

The  induced  voltage  caused  by  the  process  of  opening  the 
circuit  is  _ di 

e’=-LJt' 


(5) 

the 

(6) 


From  equation  (5)  is  obtained  the  value, 
di  _ _ aE 
dt  ( R + at )2 

aLE  aLIR 


Whence, 


«i  = 


(A  + «£)2  (R  + aty 

* Murray’s  Differential  Equations , p.  26. 


(7) 

(8) 


182 


ALTERNATING  CURRENTS 


E 

in  which  1=  — , that  is,  it  is  the  current  flowing  in  the  circuit 

R 

before  the  process  of  breaking  commenced. 

When  the  process  begins,  t = 0,  and  at  that  instant, 


(X  T ~r  LI 

e,  = — LI  = — , 
R R 


(9) 


a 


in  which  LI  represents  the  number  of  linkages  of  lines  of  mag- 
netic force  around  conductors  of  the  circuit  before  the  process 


of  breaking  the  circuit  was  begun,  and  — represents  the  ratio 

JAj 


of  the  rate  of  introducing  extra  resistance  into  the  circuit  com- 
pared with  the  initial  or  steady  resistance  of  the  circuit.  If 
the  circuit  is  broken  instantly,  that  is,  a = ao,  the  induced  vol- 
tage obviously  must  be  infinite,  as  the  formula  shows ; but  this 
fortunately  cannot  occur  in  practice,  because  the  arc  which 
occurs  at  a break  prevents  the  resistance  thereat  from  going 
instantly  to  infinity. 

If  the  line  voltage  is  not  removed  from  the  circuit  upon  the 
incipiency  of  the  break  in  the  circuit  (as  in  the  case  of  open- 
ing a switch),  it  must  be  added  to  the  self-induced  voltage  ee, 
and  the  resultant  becomes 


e — ei+E. 


The  induced  voltage,  eh  is  numerically  equal  at  time  t = 0 to 

as  many  volts  as  LI  is  a multiple  of  — . The  formula  (8)  shows 

a 

that  et  falls  off  rapidly  with  the  time,  especially  when  a is  large. 

If  the  rate  of  change  of  resistance  increases  with  t instead  of 
being  constant,  the  maximum  value  of  et  comes  when  t has  some 
finite  value  instead  of  when  t=  0. 

If  a current  of  ten  amperes  is  flowing  through  a circuit  of  10 
ohms  resistance  and  .2  henry  self-inductance,  and  the  opening 
of  a switch  gives  an  effect  at  the  initial  instant  of  introducing 
resistance  into  the  circuit  at  the  rate  of  10,000  ohms  per  second, 
the  induced  voltage  at  that  instant  is  approximately  2000  volts, 
which  is  the  maximum  value  if  the  rate  of  change  of  resistance 
does  not  increase.  If  the  drawing  out  of  the  arc  following  the 
interruption  of  metallic  contact  causes  the  apparent  rate  of  in- 
troduction of  resistance  to  increase,  the  induced  voltage  may 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  183 

continue  to  rise  to  a higher  value,  after  which  it  drops  to  zero 
on  the  breaking  of  the  arc. 

By  applying  the  same  reasoning  to  the  formula  when  resist- 
ance and  capacity  are  in  circuit  together,  it  may  be  observed 
that  ec  = — E when  t = 0 and  there  is  no  rise  of  voltage,  which 
is  to  be  expected  from  the  physical  characteristics  of  capacity. 
The  stored  energy  remains  stored  in  the  condenser.  Such 
storage  unaccompanied  by  How  of  current  is  manifestly  impos- 
sible in  the  case  of  electromagnetic  energy  except  through  some 
exhibition  of  the  phenomenon  of  coercive  force  in  an  iron  core 
or  the  like. 

56.  Rate  of  Expenditure  of  Work  in  Circuit  Containing  Self- 
inductance and  Capacity ; Analogies.  — The  effect  of  self- 
inductance has  been  compared  with  the  effect  of  inertia  in 
a moving  physical  mass.  The  inertia  effects  of  water  flowing 
in  a pipe,  as  suggested  by  Faraday,  afford  instructive  analogies. 
Thus,  on  impressing  voltage  upon  a circuit  containing  resist- 
ance and  self-inductance,  the  current  does  not  rise  to  its  full 
value  instantly,  but  increases  as  a logarithmic  function,  the  con- 
stant of  which  depends  upon  the  relation  of  the  self-inductance 
to  the  resistance  of  the  circuit.  In  the  same  way,  if  pressure  is 
exerted  upon  water  which  fills  a pipe,  the  water  cannot  begin  its 
full  flow  instantly,  on  account  of  inertia.  If  a gate  is  suddenly 
closed  in  the  pipe  after  the  flow  is  fully  under  way,  the  momen- 
tum of  the  liquid  tends  to  continue  the  flow,  and  the  gate  suf- 
fers a severe  blow.  In  the  same  way,  upon  opening  an  electric 
circuit  a bright  spark  passes  on  account  of  the  so-called  extra 
current  caused  by  the  tendency  of  self-inductance  to  uphold  the 
flow.  It  must  always  be  remembered  that  the  analogies  be- 
tween the  flow  of  electric  current  and  moving  solids  or  liquids 
are  by  no  means  exact.  For  instance,  there  is  a marked  differ- 
ence between  the  effect  of  bends  on  the  inertia  effect  in  the 
pipe  containing  water  and  in  the  electric  circuit.  Thus,  in  the 
electric  circuit,  a solenoid  has  much  more  self-inductance  than 
has  the  same  wire  straightened  out.  On  the  other  hand,  bends 
in  a water  pipe  cause  the  inertia  effect  to  be  absorbed  by 
friction.  Notwithstanding  the  differences,  the  analogies  are 
quite  useful  in  fixing  the  meaning  of  the  phenomena  and 
worthy  of  further  consideration. 

When  water  in  a pipe  is  set  in  motion,  a part  of  the  force 


184 


ALTERNATING  CURRENTS 


exerted  upon  it  at  any  moment  is  utilized  in  overcoming  the  re- 
Mdv 

action,  , caused  by  the  inertia  of  the  accelerating  mass,  and 


the  remainder  in  overcoming  frictional  resistances  (J.v). 
That  is. 


F = Av  + 


Mdv 


where  F is  the  pressure  exerted,  v the  instantaneous  rate  of  flow 
(velocity)  of  the  water,  M is  its  mass,  and  A is  a constant.  It 
is  here  assumed  that  the  frictional  resistance  is  proportional  to 
the  velocity,  which  is  true  only  when  v is  small.  When  the 
velocity  of  the  water  has  become  so  great  that  Avx  = F.  where 
vx  is  the  final  velocity,  the  acceleration  ceases,  and  the  water 
continues  to  flow  at  a uniform  velocity  vx  as  long  as  the  force 
is  applied. 

In  the  case  of  the  electric  circuit,  tlie  impressed  voltage  is 
expended  in  overcoming  the  counter- voltage  due  to  self-induct- 


Ldi 

ance  and  in  causing  the  current  to  flow  through  the  resist- 


ance R of  the  circuit,  or 


E=iR  + 


Ldi 

dt 


This  is  similar  to  the  expression  for  the  flow  of  a liquid  as 
given  above.  The  voltage  exerted  in  overcoming  the  electric 
resistance  or  electric  friction  of  the  conductor  is  represented  by 

iR,  and  represents  the  voltage  exerted  in  stor- 

ing energy  in  the  magnetic  field ; that  is,  in  changing  the 
electro-magnetic  momentum  of  the  magnetic  field.*  When  the 
current  reaches  such  a value  that  iR  = E , the  last  term  dis- 
appears and  the  current  becomes  constant.  The  expression 

Mdv^  f __  d(Mv)\  the  formuia  relating  to  the  flow  of  water 
dt  \ dt  J 

represents,  of  course,  the  pressure  or  force  exerted  in  storing 
energy  in  the  water  by  increasing  its  momentum. 

The  power  expended  in  the  electric  circuit  at  any  instant  is 


Ei  = i2  R + 


Lidi 

~1T' 


* Art.  47. 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  185 


in  which  PR  is  the  power  expended  in  heating  the  conductor 
di 

and  Li — is  the  power  expended  in  storing  energy  in  the  mag* 
dt 

netic  field.  In  this  discussion,  it  is  assumed  that  the  electrical 
resistance  of  a conductor  is  a constant,  and  is  the  same  for  a 
variable  current  as  for  a constant  one.  This  is  correct  within 
practical  limits,  provided  the  rate  of  variation  of  the  current  is 
not  too  great  and  the  conductor  is  not  too  thick. 

The  capacity  effect  in  a circuit  has  analogies  with  the  effect 
of  the  physical  capacity  of  a system  of  water  pipes  upon  the 
flow  of  water ; or,  to  make  the  comparison  more  simple,  suppose 
that  the  pipes  are  so  small  as  to  have  an  inappreciable  capacity 
but  lead  to  the  bottom  of  a large  tank.  The  force  acting  upon 
the  water  must  then  accomplish  two  purposes ; first,  it  must 
force  the  water  through  the  resistance  of  the  connecting  pipes, 
and  second,  it  must  raise  the  level  or  potential  of  the  water  to 
the  tank  level.  A formula  may  be  made  to  express  this  as  follows, 

F = Av  +JI—  Av  + 

where  H is  the  head  of  the  water  in  the  tank,  Gr  is  the  volume 
of  the  water  in  the  tank,  and  Y is  the  capacity  of  the  tank  in 
units  of  volume  per  unit  of  height  or  head.  In  case  of  an  elec- 
tric circuit  the  analogous  equation  is, 

F=Ri  + ±. 


The  power  which  is  used  can  immediately  be  obtained  by 
multiplying  the  hydraulic  equation  by  the  rate  of  flow  of  the 
water  and  the  electric  equation  by  the  current.  The  equation 
of  power  in  the  electric  circuit  is  then 

Fi  = PR  + il, 

but  as  i = — 

dt 

iF=RP  + %&- 

Cdt 

In  case  there  are  both  capacity  and  self-inductance  in  series 
in  the  circuit,  the  two  equations  must  be  combined  and 


F = Ri  + 


Ldi  q 


* Art.  52  c. 


186 


ALTERNATING  CURRENTS 


and  the  power  used  is 


iR=Ri 2 + ^ + 

dt  Cdt 


It  must  be  remembered  in  considering  this  equation  that  energy 
is  absorbed  by  the  circuit  on  account  of  the  second  term  on  the 
right  only  while  the  current  is  rising ; when  the  current  falls 
the  magnetic  field  gives  up  energy  which  is  returned  to  the 
source.  Likewise,  energy  is  absorbed  by  the  circuit  on  account 
of  the  third  term  of  the  right-hand  member  of  the  equation 
only  while  the  voltage  impressed  on  the  capacity  is  rising  and 
the  charge  in  the  condenser  is  therefore  increasing;  when  the 
charge  falls  the  condenser  gives  up  energy  which  is  returned  to 
the  source.  These  reactions  are  analogous  to  the  effects  in 
water  pipes  where  the  inertia  absorbs  energy  while  the  velocity 
is  rising,  to  return  it  as  the  velocity  falls ; while  the  tank 
stores  energy  while  being  charged  and  returns  it  on  discharge. 
The  resistance  and  iron  and  dielectric  losses  are  analogous  to 
the  frictional  resistances  in  the  pipes  and  tank. 

57.  Effect,  on  the  Transient  State  in  a Circuit,  of  Self-induct- 
ance and  Capacity  Combined.  — When  an  inductive  coil  of  con- 
, stant  inductance  L is  included  in  a 

steady-current  circuit  of  resistance  R. 
and  a condenser  of  capacity  C is  shunted 
across  a portion  of  the  circuit  of  resist- 
ance r,  as  in  Fig.  129,  the  following 
conditions  obtain  : 

The  condenser  is  charged  with  a 
quantity  of  electricit)*,  Q = CIr , where 
I is  the  steady  value  of  the  current. 
Fig.  129. -Circuit  containing  Now  if  the  impressed  voltage  is  sud- 
Resistance,  Self-inductance,  denly  removed  by  throwing  the  switch 
and  Capacity.  shown  in  the  figure  to  the  horizontal 

position,  the  condenser  will  discharge  and  the  quantity  of  elec- 
tricity which  will  pass  from  the  condenser  through  the  part  of 
the  circuit  beyond  its  terminals  is 


H 


CIr 2 
R 


At  the  same  time  the  self-inductance  will  cause  a quantity  of 
electricity  to  be  transferred  through  the  circuit  in  the  opposite 
direction,  which  is  equal  to 


1 = 


LI 
R ‘ 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  187 

Hence,  the  total  quantity  of  electricity  transferred  through  the 
circuit  is  r 

qi ~ <1c  = r^l~ 

and  the  effect  of  the  condenser  is  to  apparently  reduce  the  self- 
inductance by  an  amount  equal  to  the  capacity  of  the  condenser 
multiplied  by  the  square  of  the  resistance  around  which  it  is 
shunted.  With  iron  in  the  circuit,  L varies,  but  the  general 
relations  remain  the  same. 

The  foregoing  relates  to  the  transient  state  upon  removing 
the  impressed  voltage  which  sets  up  the  steady  current  I in  the 
circuit,  but  it  is  obvious  that  similar  conditions  are  produced 
during  the  transient  state  accompanying  the  establishment  of 
the  current  upon  introducing  the  impressed  voltage.  If  Cr2  is 
equal  to  L , it  is  obvious  that  the  charging  current  neutralizes 
the  extra  current  of  self-induction  and  the  total  circuit  acts 
like  a circuit  containing  resistance  R but  lacking  self-induct- 
ance  or  capacity.  The  neutralization  of  the  effect  of  self- 
inductance cannot  ordinarily  be  complete  on  account  of 
hysteresis  in  the  iron  core  of  the  coil  (if  an  iron  core  is  used) 
and  the  dielectric  hysteresis  of  the  condenser,  which  cause  the 
curves  of  discharge  from  the  magnetic  field  and  condenser  to 
vary  from  the  logarithmic  form. 

58.  Conditions  of  Establishment  and  Termination  of  Current  in 
a Circuit  containing  Resistance,  Inductance,  and  Capacity  in 
Series.  — a.  Current  and  Charge  under  Constant  Impressed  Vol- 
tage.— The  equation  of  voltage  when  current  is  established  by 
introducing  constant  voltage  E into  the  circuit  is 


E = Ri  + ^l+Z 

dt  C 

=Bi+Lft +hfidt 


From  (1)  we  obtain 

0 — ^ , 1 dq 

~dC  + Ld^+LCdt' 

Putting  q = 

the  equation  takes  the  characteristic  form 

R x 2 , a; 


188 


ALTERNATING  CURRENTS 


the  roots  of  which  are 


x-,  = — 


A 

2 L 


^ VI  T%  TO 


I R2 

i 

k 4 L2 

LC ’ 

1 

xs  = °- 


The  solution  of  such  an  equation  as  (2)  is  * 

q = cqe*1*  + a2er2*  + a3,  (4) 

in  which  x1  and  x2  are  the  aforesaid  roots  of  equation  (3)  and 
and  a3  are  constants  introduced  by  integration.  Also,  since 


tty,  &2P 


;=A, 

dt 


i = + a2x2er-t.  (5) 

These  equations  (4)  and  (5)  for  charge  and  current  assume  three 
forms  according  to  the  values  of  the  constants  of  the  circuit, 

Z?2  1 7?2  1 

(3)  when 


namely:  (l)whenA>A;  (2)  when  A ; 


R2  1 

= — under  which  conditions  the  roots  x,  and  xn  are 

4 A2  LG  12 

respectively  real,  imaginary,  and  equal. 

R2  1 

Case  (1).  When  — — > — — - ; 1 Yon-oscillatory  Effect.  — Under 
4 _£r  L (J 

these  conditions  the  values  of  xx  and  x2  in  equations  (4)  and  (5) 
are  real.  The  values  of  the  constants  av  a2,  and  a3  may  be 
determined  by  solving  when  t is  given  the  special  values  of  0 
and  oo. 

For  the  conditions  here  named,  q = 0 and  i=  0 when  f=0; 
q = EC  — Q and  i = 0 when  t = oo.  Substituting  these  two  sets 
of  values  successively  in  (4)  and  (5)  gives  the  following  four 
equations  : ^ + a2  + a3  = 0, 

a1x1  + a2:r2  = 0, 

Q — a3  = aff1"11  + a.ff-"0, 

4-  a2x.2enr  = 0, 

Qx 2 


from  which, 


a.  = 


xi  ~ X2 

- Qx\ 


a3  = Q. 

* Murray’s  Differential  Equations,  p.  64. 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  189 


Substituting  these  values  of  the  constants  in  (4)  and  (5),  we 
obtain 

Qx 2^  Qx  ^ 

Q = U 

l nr  nr  nr  nr 

1 x2  1 •*  2 


= Q- 


Q 


and  i = 


i = 


/ R 2 

1 

^-iL2 

LC 

( R 

U L 

Qx^c2Clt 

Q 

~ x1-x2 

X 

II 

R 

nJR2 

L 

” * 4 

C 

R + - 

1C 

IL  + 

'4  L2 

1 R2 

1 ' 

'4  L2 

LC 

V (6) 


X,  — xn 


„ V2Z  4Z2  iW  „ V2  L 4 L-  LC) 


• (7) 


These  equations  (6)  and  (7)  give  the  values  of  the  charge 
and  current  in  a series  circuit  containing  R , Z,  and  (7,  when 
R2  1 

■ > , at  any  time  t after  the  introduction  of  a constant 

4 L2  LG  J 

voltage  R in  the  circuit,  Q being  the  final  charge  corresponding 
to  the  voltage  R. 

7?2  1 

Case  (2).  When  ■ — — < — — ■;  Oscillatory  Rffect. — Under  these 
4 1 / L C 

conditions  the  roots  xx  and  x2  are  imaginary.  Hence,  using  the 
operator  j to  indicate  the  imaginary  unit  V — 1,  we  have 


Substituting 
where  a = — 


L. 

2 L J ^ LC 

R2 

4 Z2’ 

U i\\  1 - 

R2 

2 L J ' LC 

4 L2' 

Xj  = a +jb  and 

1 1 

2 L V 

LC 

and  are  real  quantities, 


190 


ALTERNATING  CURRENTS 


equations  (4)  and  (5)  give  * 

q = a1er,<  + a 2G4  + a3 
= alGa+jb)t  + a2e^b)t  + a3 
= eat(A'  cos  bt  + B'  sin  bt)  + a3 
= A'eat  cos  bt  + B' eat  sin  bt  + a3, 
in  which  A!  — ax  + a2  and  B'  = j (a1  — a2). 

Also  i — (A! a + B'b)eat  cos  bt  + (B1  a — A'b)Gt  sin  bt. 

If  p is  an  angle  of  which  tan  p = — , it  follows  that 

n 

m n 

sin  p — — — cos  p = — . 

's/m1  + n2  V m2  + n2 


and  m cos  a + n sin  u = Vm2  + n2  sin  (a  + p) . 

Therefore, 

q — eal  a/A'2  + B 12  sin  (bt  + 0)  + a3,  (8) 

i = ea'  V (A'a  + B'b ) 2 + ( B'a  — A'b ) 2 sin  (it  + 0'),  (9) 


m 


i • , a , _id'  , /i,  i (A/a  + 15'i) 

which  0 = tan  1 — , and  0'  = tan  1 1 

B1  (B'a -A'b) 


The  values  of  the  constants  of  the  equations  may  he  deter- 
mined as  in  the  previous  case  by  solving  when  t is  given  the 
limiting  values  of  0 and  oo.  As  before,  q — 0 and  z = 0 when 
t — 0 ; q = Q and  i = 0 when  t = go.  Whence, 


A'=-Q, 
B'=-A'?  = 


~QR 


2 L 


q = Q-Qe  2L 


a3  = Q. 


\ L 0 


I 1 R2 
LG  4 L2 


x Sill 


LG\ 


J 1 

R2 

'LG 

4 L2 

and  (9),  we 

"Jl  _ 

R2 

^ LG 

4 Z2 

• <+  e 


(10; 


See  Art.  71  and  Murray’s  Differential  Equations,  p.  67. 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  191 


and 


i = e 2L- 


E 


■ I 1 

'^LC  4 


R?_ 

I? 


. r i e2 

Sm'X(7  4 L2 


(11) 


in  which  6 — tan  1 


2 lJE-E 

\ TO  1 T 


LG  4 L2 

R 


These  equations  (10)  and  (11)  give  the  values  of  the  charge 
and  current  in  a series  circuit  containing  R , L , and  C,  when 
R2  1 

— — < — — , at  any  time  £ after  the  introduction  of  the  constant 
4 Ir  LC 

voltage  E in  the  circuit.  It  will  be  noted  that  the  expressions 
for  charge  and  current  each  contain  the  product  of  a logarith- 
mic term  and  a sine  term,  and  their  values  are  recurrent 
functions  of  time  of  an  oscillatory  nature.  The  value  of  the 
charge  at  any  instant  is  equal  to  the  difference  between  a con- 
stant term,  Q = CE,  and  an  oscillatory  term  which  is  in  the  lead 


1 

R2 

*LC 

4 I? 

R 

of  the  current  by  an  angle  6 of  which  tan  6 = 

The  current  consists  of  a sinusoid  modified  by  a vanishing 
logarithmic  curve,  and  its  oscillations  are  of  gradually  vanish- 
ing amplitude. 

R2  1 

Case  (3).  When  — — = — Under  this  critical  condition, 

4 L2  LC 

the  roots  xx  and  x2  of  the  equation  (3)  become  equal,  or 

R 1 


X,  = X„  = X = — — — 


2 L 


VLC 


and  the  solution  becomes  : 

q — a^C1  + + az , 

i = axxeri-{-  azxtext  4- 


(12) 

(13) 


The  values  of  the  constants  may  be  determined  by  solving 
when  t is  given  the  special  values  of  0 and  oo.  As  before, 
q = 0 and  * = 0 when  t = 0 ; and  q = Q and  i = 0 when  t = oo. 

Then,  ax  = — Q, 


*See  Murray’s  Differential  Equations , p.  65. 


192 


ALTERNATING  CURRENTS 


R 


«2  = - Q = - — = 


Q 


2L 


V LO 


a3  — Q‘ 


Substituting  these  values  in  equations  (12)  and  (13)  we 
obtain, 

Rt\  " 


q = Q~  (/  I + )e  2 £, 


2 A 


7?2  E1  /« 

i=QjL2te^  = -t^. 


(14) 

(15) 


These  equations  (14)  and  (15)  give  the  values  of  the  charge 
and  current  in  a series  circuit  containing  R,  L , and  (7,  when 

R2  1 

= — at  any  time  t after  the  introduction  of  a constant 

4 A2  LC 

voltage  E in  the  circuit.  This  condition  affords  the  most  rapid 
rate  of  charging  for  the  circuit.  A decrease  of  R would  put 
the  circuit  in  an  oscillatory  condition,  and  an  increase  of  R 
would  put  the  circuit  in  a condition  represented  by  equations 
(6)  and  (7). 

b.  Current  and  Charge  on  Withdrawal  of  Impressed  Voltage. — 
On  removal  of  the  impressed  voltage  from  a circuit  containing 
R,  L , and  C in  series,  the  equation  of  the  circuit  is 


= m+i§+c. 


ir* 


idt 


- Vd(i  | | ? 

~ dt+  dt 2 +c' 

Putting  q = €**, 

the  equation  takes  the  characteristic  form 


the  roots  of  which  are 


(16) 


+ LX  + 

1 

LC 

0, 

lR2 

1 

2 A + ^ 

4 A2 

LC' 

R 

I A'2 

1 

2 L ^ 

“4  A2 

LC 

(17) 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  193 


The  solution  takes  three  forms  according  as  these  roots  are  real, 
imaginary,  or  equal,  or  when 

...  1 , JR*  1 /QN  1 R 2 1 

(1)  Tl^lo’  (2)  when  (3)  when  4I?  = LC' 

R2  1 

Case  (1).  When— — >— — ; Non  - oh  dilatory  Effe  ct.  — Under 
T J-J  -Lj  0 

these  conditions  the  solution  of  equation  (16)  takes  the  form 

q = aqe*1*  + (18) 

i — axx1er't  + a2x2eXit.  (19) 

To  determine  the  constants  we  have,  when  t = 0,  q = Q and 
i = 0 ; and  therefore 


Qz* 


_ Qx-i 


‘X'i  OCn 


Substituting  these  values  in  (18)  and  (19),  we  obtain 
R 


q=Q 


4 Lxl* 

1 

LC 

and 


i = ■ 


-Q 


Q 


4 L' 


2 LC 


r 

+ q 

1 

i 

R 

R2 

i 

4 L2 

LC 

— (IL+  -\/ 

P \-2L  v 4L* 

-(A+VWIT\ 

: \2L+  V 4£2  LC) 


'll2  LC 


(20) 


(21) 


These  equations  (20)  and  (21)  give  the  values  of  the  charge 
and  current  in  a series  circuit  containing  R , _Zi,  and  U,  when 
R2  l 

> , any  time  t after  the  withdrawal  of  the  impressed 

4 U LC 
voltage  E. 

Case  (2).  When  ; 

conditions  the  roots  xx  and  x2  of  equation  (17)  are  imaginary 
and  proceeding  as  in  case  (a.  2),  p.  189,  the  equations  for  charge 
and  current  become 


Oscillatory  Effect.  — Under  these 


q ---  eaty/ fA1')2  + ( B1')2  sin  ( bt  + d), 


o 


(22) 


194 


ALTERNATING  CURRENTS 


i=ealV(Ara  + B'b)2  + (B'a-A'by  sin  (bt  + 0'),  (23) 


where 


R , ^ 1 

a = , o — \ 

9 T V 


i(7 


it*2 
4 X2’ 


, n .d/  -i  , /w  -I-  R b 

tan  6 = — , and.  tan  0 = — — - — — . 

B ' B'a  - A'b 

To  determine  the  values  of  the  constants  in  (22)  and  (23) 
we  have,  when  t = 0,  q = Q and  i = 0. 


Then, 


A'  = Q and  B'  = - -A'  = - ® Q. 

b b 


Substituting  these  values  in  (22)  and  (23),  we  obtain, 

" vzv 


q = Qe 


2 L 


LG 


W 


R 2 

LG  4 i2 


x 


and 


sm 


i = — e 


'•LO 


Rt 

1L 


Q 


t + tan 


-l 


2ijxi^r 

4 L2 


R 


LG 


w 


i2 


sin 


( Vi- 


i?2 


LG  4i2 


(24) 

(25) 


1 LG  4 X2 

These  equations  (24)  and  (25)  give  the  values  of  the  charge 
and  current  in  a series  circuit  containing  R,  L , and  (7,  when 
R2  1 

- — - < — — , at  any  time  f after  the  withdrawal  of  the  impressed 

voltage  E.  It  will  be  noted  that  the  discharging  process  is  of 
an  oscillatory  nature,  the  expressions  for  q and  i being  each 
made  up  of  the  product  of  a logarithmic  term  and  a sine  term, 
and  that  the  charge  leads  the  current  by  an  angle  of  which 


tan  6 = 


2 lJ—  - 

'LG 


R2 
4 L2. 


R 


The  period  of  each  oscillation  is 


z TV 


1 1 _ 

R2 

JLG 

4 L2 

which  is  the  natural  oscillatory  period  of  the  circuit.  If  R is 
small  compared  with  L , this  is  approximately  equal  to  2 7rx  LG. 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  195 


R 2 l 

When  — Under  these  conditions  the 

4 U LC 


Case  (3). 

roots  of  equation  (17)  become  equal,  or 


X,  = x0  = x = — 


B 


-1 


2 L y/LC 

and  the  solution  of  equation  (16)  becomes 

q — axCl  + a^e*1, 

and  i = a^x^1  + a2xteTt  + a2€Tt. 


(26) 

(27) 


To  determine  the  values  of  the  constants,  we  have  q — Q and 
* = 0 when  t = 0. 


Hence, 


ax  = Q and  a2  = — Qx  = 


Substituting  these  values  in  (26)  and  (27),  we  obtain 


1? 

i = -~te  2L. 
-Lj 


Rt 

2L 


(28) 


(29) 


These  equations  (28)  and  (29)  give  the  values  of  the  charge 
and  current  in  a series  circuit  containing  R , L , and  C,  when 
B2  1 ■ 

= — - at  any  time  t after  the  withdrawal  of  the  impressed 

4Z2  LC  J * 

voltage  j E from  the  circuit.  This  relation  of  B , X,  and  C affords 

the  most  rapid  discharge  of  the  circuit.  A decrease  of  R would 

throw  the  discharging  process  into  the  oscillatory  state,  and  an 

increase  of  R would  put  the  circuit  in  a condition  represented 

by  equations  (20)  and  (21). 

c.  Current  and  Charge  when  the  Impressed  Voltage  is  a Sine 
Function  of  the  Time.  — In  this  case  the  instantaneous  voltage 
conditions  are  represented  by 

e = em  sin  wt  = iR  4-  + 7 

dt  C 


dt  CC  dt 

em  . , cPq  R dq  q 

L Sin  = dp  + L dt  + LC 


or 


(30) 


196 


ALTERNATING  CURRENTS 


o ^ 

In  this  case  of  a sine  function,  u>  is  equal  to  ~jT=  2 the 

angular  velocity  of  a rotating  vector  generating  the  sinusoid. 

The  above  is  a linear  differential  equation  of  the  second 
order  which  is  to  be  solved  to  find  the  values  of  q and  i in 
terms  of  em,  R , L,  (7,/,  and  t.  The  solution  of  such  an  equation 
consists  of  the  sum  of  two  parts,  the  complementary  function 
and  the  particular  integral.  The  complementary  function  is 
the  integral  obtained  by  equating  the  second  member  to  zero. 
In  other  words,  it  is  the  solution  of  case  (5)  ante , i.e.  when  e = 0, 
and  takes  the  form  * 

q = a1eT,t  + a2eT2\ 


i = a^x-^e1'1  + a2x2eXlt. 

The  particular  integral  contains  no  arbitrary  constants  and  is 
equal  to  f 


9 = 


R 


D1*  + — D + — — 
L LG 


— •••  ~ sin  a>t  = - — % sin  cot 

1 L I)  — a R 


^ m ,a(  i c—at 


R 


/* 


— e„ 


and 


tan  8 = 


+ 

> 

3 

1 

e 

bn 

1 

lb 

tc 

co*L 

1 

Cl  ■ — 

R 

OR' 

e 

bi 

1 

e u 

coR 

R 

-cos  (cot  — d). 


The  complete  solution  is  therefore 

9 = Pm  — — cos  (cot  - g)  + u1eri/  + a2e*1.  (31) 

+ (»£—) 

In  a similar  way  the  complete  solution  for  the  current  becomes 
i — — e'n  sin  (cot  — 8)  + a,x,erit  + u0rroer!/.  (3d) 

V^+(«,z-L) 

* Murray’s  Differential  Equations,  Chap.  VI. 
f Murray’s  Differential  Equations,  Arts.  58  and  62. 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  197 


These  equations  give  the  values  of  the  current  and  charge  at 
any  time  t after  impressing  the  voltage  in  a series  circuit  con- 
taining constant  it,  i,  and  (7,  when  such  voltage  is  sinusoidal. 
The  above  equation  for  current  may  be  written 


i = , . €"’  ~ — -sin  (a-0)  + a1x1ecit  + a2x2ex*t.  (33) 

The  angle  6 is  the  angular  difference  of  phase  between  the  cur- 
rent in  the  circuit  and  the  impressed  voltage  ; that  is,  it  is 
the  angle  of  lag  (or  lead)  of  the  current  for  frequency/. 

Neglecting  the  exponential  terms,  which  are  rapidly  evanes- 
cent where  xx  and  x2  are  real,  the  equation  for  instantaneous 
current  under  a sinusoidal  voltage  is 


where  em  is  the  maximum  voltage  and  the  angle  6 is  either  posi- 
tive or  negative  according  as  capacity  or  self-inductance  pre- 
dominates. The  current  is  evidently  a maximum  when  sin 
(a—  0)  = 1,  and  therefore 


The  relation  of  effective  current  and  voltage  can  be  obtained 
b}7  dividing  both  sides  by  V 2,  and  then 


I — 


E 


\/«2 


2 + 2 7T.fi  - 


2 7 tfC, 


The  term  yjR2  + (ZirfL  — --  is  called  the  Impedance  of 
\ 2 ttj  G y 

the  circuit  and  is  composed  of  the  square  root  of  the  sum  of  the 
squares  of  two  quantities,  the  first  of  which,  R,  is  the  resistance 

of  the  circuit  and  the  other  yl^fL  — ■ ) is  called  the  React- 

\ ‘ 2 irfCJ 


ance  of  the  circuit. 


198 


ALTERNATING  CURRENTS 


If  L is  zero  and  C is  infinite,  the  equation  (38)  reduces  to  the 

value  _ g 

i = V sin  «, 

R 

as  already  derived  for  the  current  flow  when  a sinusoidal  vol- 
tage is  impressed  on  a circuit  containing  resistance  alone. 

If  C is  infinite  (which  is  equivalent  to  saying  that  no  capacity 
voltage  arises  in  the  series  circuit),  but  R and  L have  finite 
values,  the  equation  reduces  to  that  on  page  148,  or 


ri 


i — 


■ sin  (a  — 6')  + Axe  z. 


4-  (2  77 -fLy 

The  value  of  Ax  may  be  derived  by  giving  a and  t the  values 
and  tv  corresponding  to  the  instant  of  introducing  the  voltage 
in  the  circuit,  at  which  instant  i = 0 and  the  second  term  of 
the  right-hand  member  must  be  equal  to  but  opposite  the  first 
term ; 


whence 


Ax  = - 


Rti 

e L sin  (oq- 


■n 


V R 2 + (2  tt/L)2 

which  is  constant,  because  tx  and  «1  and  O'  have  particular  values, 

and  Axe  l = _ 


-2lL=hl  . 

z sin  («j  — 6 '). 


Vi?2  + (2  tt/L)2 
The  value  of  t is  measured  from  the  instant  when  the 
sinusoidal  voltage  passes  through  the  preceding  zero  from 
negative  toward  positive  values ; and  tx  is  the  value  of  t at  the 
instant  of  switching  the  voltage  into  the  circuit. 

If  L is  zero,  but  R and  C have  finite  values,  the  equation 
reduces  to  that  on  page  174,  or 

e’"  — sin  (a  + 6"')  4-  Bxe 


i = 


_t_ 
RC . 


1 Y 

2 77-/CV 


The  value  of  Bv  derived  from  giving  a and  t the  values  ax 
and  tv  corresponding  to  the  instant  of  introducing  the  voltage 
in  the  circuit,  is 


Bx=- 


VR2  + 


Bxe  nc  — _ 


l Y 

2 77 fC) 

n 


V!+ 


— T 

2 irfCJ 


eRC  sin  («j  + 6 


e Rc  sin  (aj  4-  0"'). 


and 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  199 


Fig.  130.  — Effect  of  switching-in  on  Circuit  containing  R and  L. 


200 


ALTERNATING  CURRENTS 


59.  Effect  of  Exponential  Terms.  — The  exponential  members 
of  the  equations  of  case  ( c ) show  the  transient  effect  on  the  form 
of  the  current  wave  which  occurs  upon  first  impressing  sine 
voltage  on  a circuit  containing  resistance  and  inductance,  resist- 
ance and  capacity,  or  resistance,  self-inductance,  and  capacity, 
in  series.  In  circuits  with  resistance  and  inductance  or  resist- 
ance and  capacity  the  exponential  term  quickly  reduces  to  zero 
and  can  be  then  neglected.  The  actual  influence  of  the  expo- 
nential term  in  either  of  these  two  cases  is  to  distort  the  current 
wave  for  the  earlier  periods,  and  the  extent  of  the  effect 
depends  on  the  value  of  (rq  — d)  at  the  instant  of  introdu- 
cing the  voltage  into  the  circuit.  The  value  of  a1  expresses 
the  advance  of  the  voltage  at  the  instant  of  its  introduction  in 
the  circuit,  and  the  value  of  (cq  — d)  expresses  the  advance 
of  the  corresponding  normal  current.  Consequently,  when 
eq  = d,  the  exponential  value  is  zero  and  the  current  starts  from 
zero,  following  a regular  sine  wave  ; and  when  cq  = 90°  -f  d,  the 
exponential  obtains  its  largest  value  and  the  current  wave  is 
given  the  maximum  of  distortion.  Figures  130  and  131  show 
the  form  of  current  wave  for  the  first  few  cycles  where  a sinu- 
soidal voltage  of  100  volts  is  switched  onto  a circuit  containing 
5 ohms  and  .1  henry  and  on  a circuit  containing  100  ohms  and 
100  microfarads,  the  frequency  being  60  periods  per  second  in 
each  instance.  The  curves  are  shown  for  cq  = d,  cq  =45°  + d, 
cq  = 90°  + d,  <q  = 135°+  d. 

In  Fig.  130  a the  sinusoid  marked  E shows  the  position  of 
the  voltage  ; the  sinusoid  marked  I shows  the  current.  The 
current  grows  up  in  the  circuit  without  deviation  from  the  true 
sine  form,  since  the  switch  is  closed  at  the  instant  when  « = d ; 
namely,  at  the  instant  when  the  sine  current  would  normally 
pass  through  zero,  and  the  exponential  term  of  the  equation  is 
therefore  zero.  In  Figs.  130  b , c,  and  d,  the  voltage  curve  has 
been  omitted.  The  dotted  curve  represents  the  normal  sine 
form  of  the  current,  corresponding  to  the  curve  I of  Fig.  130  a, 
and  the  full  line  curves  represent  respectively  the  exponential 
term  (which  is  equal  and  opposite  to  the  value  of  the  normal 
sine  curve  at  the  instant  of  switching  in)  and  the  actual  current 
flow.  The  latter  is  distorted  for  several  periods  from  the  nor- 
mal sine  form  by  the  effect  of  the  reactance  in  the  circuit. 

Figures  131  a,b,.c,  and  d similarly  represent  the  relations 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  201 


when  the  sinusoidal  voltage  is  switched  on  a 


circuit  of  fixed 


/ ~ \ 
1 1 \ I 


Jh< 

T 


capacity  and  resistance. 

The  effects  of  switch- 
ing a sinusoidal  voltage 
onto  a circuit  of  fixed 
resistance  and  self-in- 
ductance or  a circuit  of 
fixed  resistance  and  ca- 
pacity are  thus  made 
plain.  In  case  the  self- 
inductance varies  much 
during  the  cycles,  as  in 
the  case  of  a coil  with 
an  iron  core  that  be- 
comes highly  saturated 
in  each  half  cycle,  or 
the  capacity  varies  dur- 
ing the  cycles,  as  in  the 
case  of  certain  liquid 
condensers,  the  expo- 
nential terms  become 
very  complex  on  account 
of  the  variation  of  L or 
(7,  and  the  magnitude  of 
the  first  rush  of  current 
may  become  serious. 

The  distortion  of  the 
form  of  the  current 
wave  during  the  first 
few  cycles  may  then  be 
much  greater  than  is 
indicated  in  Figs.  130 
and  131. 

When  the  voltage  is 
withdrawn  from  the  cir- 
cuit without  changing 
the  constants  of  the 

circuit,  it  is  obvious  that  the  first  terms  of  the  equations  reduce 
to  zero,  and  the  current  dies  down  along  an  exponential  curve 
from  its  value  at  the  instant  of  withdrawal  of  the  voltage. 


= 90°+ 0° 


. 

/ 

o 

© 

— T"  T d 

a,  = 135°+0° 

Fig.  131.  — Effect  of  switching-in  on  Circuit  con- 
taining Ii  and  C. 


202 


ALTERNATING  CURRENTS 


On  the  other  hand,  if  the  voltage  is  withdrawn  in  the  usual 
manner  of  opening  the  circuit,  the  value  of  R is  rapidly  changed 
to  infinity.  An  inspection  of  the  two  exponential  terms  shows 
that  this  may  cause  a great  disturbance  in  an  inductive  circuit 
during  the  discharge  of  the  magnetic  field ; but  the  current 
merely  ceases  in  the  circuit  with  capacity,  leaving  the  con- 
denser charged.  In  the  latter  case,  a serious  momentary  current 
rush  and  distortion  of  the  current  wave  may  occur  if  the  vol- 
tage, when  switched  on  the  circuit  again,  chances  to  be  reversed 
in  relation  to  the  condenser  voltage. 

Turning  again  to  the  equations  representing  the  conditions 
when  resistance,  self-inductance,  and  capacity  are  all  three  in 
series,  (equations  (31)  and  (32)),  these  apply  to  all  instants  of 
time,  including  the  instant  t = tx  when  the  voltage  is  introduced 
in  the  circuit.  If  the  capacity  of  the  circuit  is  without  a 
previous  charge,  the  values  of  q and  i are,  of  course,  both  zero 
at  this  instant ; and,  solving  for  oq  and  a2  under  these  circum- 
stances gives,  when  xx  and  z2  are  real, 

em  e~Xlh  sin  (coty  — 6 4-  p') 

d . . } 

Z x2  — x1  cos  p 

_ _ em  e~x-tl  sin  (coty  — 6 + p'~) 

a2 '/  ' i ’ 

Z Xy  Xy  COS  p 

in  which  Xy  and  x2  have  the  values  heretofore  given  them,  Z is 
the  impedance  of  the  circuit  (=  yjR2  + (2  irfL  - — ^ 
ty  is  the  value  of  t at  the  instant  of  introducing  the  voltage 
into  the  circuit,  tan  p = - 2,  and  tan  p'  = Therefore, 

0)  (O 


(lyXyG^  + dyXylf2*  — 


2 zj&_ 

V4  Z2 


1 

LO 


JL  _ e - G h-  VSrS) <* - « Sin  (coty-d  + p) 

9 T.  » -t  T. 2 T.C! ) cos  p 


A 

2 L 


+ ' 


1 R2 

1 N 

*4  I? 

LGj 

1 R2 

1 ^ 

4 L2 

LG) 

V (2*  + V4 % -Ic)«- « sin  (tot  1 — d + p_'V 


cos  p 


When  the  circuit  constants  are  fixed,  the  right-hand  member 
of  the  last  equation  reduces  to  zero  rather  rapidly,  the  precise 


SELF-INDUCTION,  CAPACITY,  KEACTANCE,  AND  IMPEDANCE  203 


rate  of  decrease  as  time  grows  depending  upon  the  relative 
values  of  R,  L,  and  C ; and  the  current  will  quickly  come  to  a 
sinusoidal  value  with  relatively  little  disturbance.  However, 
if  R , L,  and  0 vary  during  the  cycles  of  current,  the  disturb- 
ance may  be  serious  on  account  of  the  instantaneous  flow  of  an 
excessive  current. 

1 R2 

When  — — = — — , equations  (31)  and  (32)  take  the  special 


LC  4 L2 


forms, 


q = — ~ cos  ( cot  — 0)  + (ax  + a2£)er( 


i 


Z 


sin  (cot  — 0)  + (ape  + a2xt  -(-  «2)  e Tt. 


Solving  for  ax  and  a2  in  these  equations  when  q = 0 and  i = 0, 
which  are  the  values  of  q and  i at  the  instant  t = tx  when  the 
sinusoidal  voltage  is  introduced  into  the  circuit,  gives 


= sin  (<»U  - d + p)e 
cos  p 


Sill 


em  . (cot,  — 6 + »') 
a„  — . — sm  - — 1 ' e xl\ 


cos  p 

in  which  tan  p = ^ ^ and  tan  p'  = -,  where  x = — 

'•*  co  2 L 


cot , 


Therefore  (apr  + a2xt  + u2)  e**  = (A  -(-  i?t)  e 2 z . 


When  < —5—  and  the  values  of  2;,  and  2-,  are  therefore 
4 L2  LC  12 

imaginary,  equations  (31)  and  (32)  take  the  forms 


q = — ^jr  cos  (cat  — 0)  -f  Meat  sin  (bt  + v ), 


i = ~ sin  (ait  — 0)  + Neat  sin  (bt  -f  v'~). 

z 

Solving  these  for  the  values  of  M,  iV",  v and  v'  when  t = tv  and 
therefore  q and  i are  both  equal  to  zero,  results  in  expressions 
affording  for  i the  equation 

p „ mt-h)  li  in 

* = J sin  (eat -O')  + Pe  2 1 Smt-\—  - 4^2 » 

in  which  P is  a real  function  of  em , R,  L , C , and  £r 


204 


ALTERNATING  CURRENTS 


These  equations  show  that  the  current  will  grow  to  its 
normal  sinusoidal  value  without  great  disturbance  after  the 
voltage  is  switched  into  circuit,  and  will  die  away  without 
great  disturbance,  if  the  values  of  X,  X,  and  C give  real  roots; 
but  if  the  roots  are  imaginary,  oscillatory  effects  (instead  of  a 
plain  exponential)  of  a frequency  fixed  by 


are  superimposed  on  the  normal  sinusoid  at  both  switching  in 
and  switching  off  the  voltage,  and  the  disturbances  may  be 
quite  great  at  either  switching  in  or  switching  off  the  voltage 
if  «j  chances  to  have  an  appropriate  value  ; especially  if  fx  is  not 
very  different  from  f,  as  is  not  unusual.  If  these  conditions 
are  accompanied  by  variable  values  of  X,  X,  or  C , the  disturb- 
ances may  be  still  more  magnified. 

Prob.  1.  A circuit  has  resistance  of  20  ohms  and  self-induct- 
ance of  .01  henry  in  series.  What  steady  voltage  impressed 
on  the  circuit  will  bring  the  current  to  a value  of  20  amperes 
at  the  instant  that  it  is  increasing  at  the  rate  of  1000  amperes 
per  second? 

Prob.  2.  How  much  power  is  being  expended  in  the  cir- 
cuit of  problem  1 at  the  instant  in  question  ? What  proportion 
of  this  power  is  being  converted  into  heat  in  the  conductor 
resistance  and  what  proportion  is  being  stored  in  the  magnetic 


Prob.  3.  A circuit  of  10  ohms  resistance  has  a capacity  of 
20  microfarads  in  series  with  the  resistance.  What  steady 
voltage  impressed  on  the  circuit  will  bring  the  current  to  a 
value  of  20  amperes  at  the  instant  when  the  charge  equals  ^ 
coulomb  ? 

Prob.  4.  How  much  power  is  being  expended  in  the  circuit 
of  problem  3 at  the  instant  in  question  ? What  proportion  of 
the  power  is  being  converted  into  heat  in  the  conductor  resist- 
ance and  what  proportion  is  being  stored  in  the  condenser  ? 
What  value  does  the  condenser  charge  finally  reach,  and  how 
much  energy  is  then  stored  in  the  charge  ? 


field  ? 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  205 

Prob.  5.  A circuit  has  a resistance  of  10  ohms,  a self-induct- 
ance of  .02  of  a henry,  and  a capacity  of  100  microfarads,  all 
in  series  relation.  At  a certain  instant  the  current  is  20 
amperes  and  is  increasing  at  the  rate  of  2000  amperes  per 
second.  What  steady  voltage  is  impressed  on  the  circuit  ? 
The  charge  in  the  condenser  at  the  instant  is  .001  of  a coulomb. 

Prob.  6.  How  much  power  is  being  expended  in  the  circuit 
of  problem  5 at  the  instant  in  question  ? What  proportion  of 
the  power  is  being  converted  into  heat  in  the  conductor  resist- 
ance, what  proportion  is  being  stored  in  the  magnetic  field, 
and  what  proportion  is  being  stored  in  the  condenser  ? What 
value  does  the  condenser  charge  finally  reach,  how  much 
energy  is  then  stored  in  it,  and  how  much  energy  is  then  stored 
in  the  magnetic  field  ? What  is  the  maximum  value  of  the 
energy  stored  in  the  magnetic  field  and  how  long  after  the 
introduction  of  the  impressed  voltage  in  circuit  is  it  reached  ? 

Prob.  7.  If  the  impressed  voltage  of  problem  5 instantane- 
ously falls  to  zero  after  a steady  state  of  condenser  charge  has 
been  reached  without  altering  the  electrical  constants  of  the 
circuit,  how  long  does  it  take  for  the  condenser  to  fully  dis- 
charge, and  what  is  the  maximum  value  of  the  discharge 
current  ? 

Prob.  8.  A circuit  has  200  ohms  resistance,  .02  henry  self- 
inductance and  5 microfarads  capacity,  in  series.  Draw  the 
curves  of  rising  and  falling  current  when  100  volts  are  steadily 
impressed  on  the  circuit  until  the  charge  rises  to  its  full  value 
and  the  impressed  voltage  is  then  withdrawn  from  the  circuit. 

Prob.  9.  Draw  the  curves  as  in  problem  8 when  the  resist- 
ance is  1 ohm,  the  self-inductance  10  henrys,  and  the  capacity 
10  microfarads. 

Prob.  10.  Draw  the  curves  as  in  the  preceding  problems 
when  the  resistance  is  1 ohm,  self-inductance  .2  of  a henry,  and 
the  capacity  200  microfarads. 

Prob.  11.  Draw  the  curves  as  in  the  preceding  problems 
when  the  resistance  is  .1  of  an  ohm,  the  self-inductance  5 
henrys,  and  the  capacity  500  microfarads. 

Prob.  12.  If  the  resistance,  inductance,  and  capacity  of  Prob. 
5 are  individually  segregated  in  the  circuit,  how  much  voltage 


206 


ALTERNATING  CURRENTS 


is  impressed  on  the  capacity,  how  much  is  impressed  on  the 
inductance,  and  how  much  is  absorbed  in  the  iR  drop,  at  the 
instant  considered  ? 


60.  The  Time  Constant  of  a Circuit.  — The  formula 


• E(  i 

shows,  as  already  stated,  that  the  theoretical  curve  represent- 
ing the  rise  or  fall  of  the  current  in  a circuit  containing  re- 
sistance and  self-inductance,  is  logarithmic.  The  formula  shows 
that  when  L is  very  small,  the  current  almost  immediately 
JE 

takes  its  full  value  I = — on  closing  the  circuit  or  falls  to  zero 

R -Rt 

when  the  voltage  is  removed,  since  e L quickly  becomes  negli- 
gible in  comparison  with  unity.  Theoretically,  when  induct- 
ance is  present,  the  current  can  rise  to  its  full  value  only  after 

-Rt 

an  infinite  time,  yet  e L becomes  practically  negligible  after 
a comparatively  short  interval. 

Since  resistance  has  the  absolute  dimensions  of  a velocity 
(a  length  divided  by  a time)  and  inductance  has  the  dimen- 
sions of  a length,  the  ratio  ~ has  the  dimensions  of  a time  ; 

R 

this  ratio,  in  the  case  of  any  circuit,  is  therefore  generally 
called  the  Time  constant  of  the  circuit,  and  may  be  represented 

by  the  Greek  letter  r.  In  the  preceding  equation,  — repre- 

R 

sents  the  value  which  the  current  would  instantly  reach  when 
under  the  constant  impressed  voltage,  were  there  no  inductance 
in  the  circuit.  This  is  the  same  as  the  ultimate  value  when 
there  is  inductance.  The  equation  may  therefore  be  written 

* = 7(1-6"). 

When  t = t,  this  becomes 

I 


i=I- 


2.718’ 


or 


* = .632  7. 


The  current  therefore  reaches  .632  of  its  ultimate  or  full 
value  in  a time  equal  to  t seconds.  The  value  of  the  time  con- 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  207 


stant  is  therefore  a measure  of  the  growth  of  the  current  in  a 
circuit  after  a fixed  voltage  is  impressed,  and  it  is  obvious  that 
in  a circuit  of  great  inductance  and  also  great  resistance,  the 
current  practically  reaches  its  full  value  as  quickly  as  in  a 
circuit  of  small  inductance  and  proportionally  small  resistance. 

When  the  circuit  is  one  containing  resistance  and  capacity 
instead  of  resistance  and  self-inductance,  the  time  constant  t is 
equal  to  R C,  and  in  this  case  it  is  a measure  of  the  growth  of 
the  charge  in  the  circuit.  When  the  circuit  contains  resist- 
ance and  capacity  and  inductance,  it  has  no  real  time  constant, 
but  the  apparent  time  constant  is  less  than  that  of  the  resist- 
ance and  inductance  or  that  of  the  resistance  and  capacity. 

Eddy  currents  evidently  reduce  the  value  of  the  time  con- 
stant by  their  distortion  of  the  curves  of  current  flow.*  A 
variable  self-inductance  due  to  magnetic  saturation  and  hys- 
teresis also  distorts  the  curves  of  current  change,  and  hence 
changes  the  time  constant.  Thus,  substituting  the  value  of 
self-inductance, 

t _ ^ rrnn'  Ag 

~ 10s  109  ’ 


in  the  formula  for  the  fall  of  current  gives 


._E  -(; 
* R e 


1 pout 

4 irnn'AfA. 


) 


i 


and  at  any  given  time  i is  proportional  to  e A similar 
condition  exists  for  rise  of  current.  As  g and  therefore  L varies 
during  the  entire  time  of  rise  or  fall  of  current  in  a circuit,  it  is 
evident  that  the  curve  cannot  be  logarithmic.  Raising  the  value 
of  i by  the  power  g gives 


/ -p\y-  _(uw\ 

j € ^4  7rW7i'*4/' • 


i can  therefore  be  obtained  for  any  instant  of  time  (tj)  if  the 
constants  of  the  energizing  coil  are  known,  provided  there  is 
no  magnetic  leakage. 


^ \R 


7tt\  r ( \ 

Jh\l,  \4  nnn’  A/  _ 


where  A is  a constant  for  the  instant  tv  or 

= A,  and  g log  q = log  S. 


* Arts.  49  and  111. 


208 


ALTERNATING  CURRENTS 


If  the  constants  of  the  circuit  and  the  magnetization  curve  for 
the  circuit  are  known,  the  value  of  /a  for  each  value  of  i can  be 
obtained.  A curve  can  therefore  be  drawn  in  which  the  ordi- 
nates are  equal  to  p.  log  i — where  /a  is  the  permeability  of  the 
circuit  when  the  current  i is  flowing  — and  the  abscissas  are 
log  i.  Log  il  is  then  the  abscissa  of  a point  on  this  curve  which 
has  an  ordinate  n log  iv  and  il  can  be  obtained  from  logarithmic 
tables.  In  this  manner  any  number  of  values  of  the  rising  or 
falling  current  can  be  obtained  with  due  account  of  changes  in 
permeability  caused  by  the  phenomena  of  saturation  and  hyster- 
esis. The  curve  of  current  will  contain  harmonics  caused  by 
the  phenomena  of  saturation  and  hysteresis. 

61.  Examples  of  Time  Constants.  — The  following  are  the 
time  constants  of  some  of  the  circuits  for  which  inductances 
have  previously  been  given.*  A Wheatstone  bridge  resistance, 
when  properly  wound,  generally  has  a time  constant  of  a 
millionth  of  a second  or  less;  electric  bell,  4.8  millionths  of  a 
second;  telephone  call  bell,  nearly  .02  of  a second;  modern 
short-core  telephone  bell,  about  .006  of  a second;  armature  of 
a small  magneto  generator,  from  .005  to  .013  of  a second; 
telephone  receiver,  with  diaphram,  about  .001  of  a second; 
modern  bipolar  telephone  receiver,  .0005  of  a second;  mirror 
galvanometer  for  marine  signaling,  .0016  of  a second;  mirror 
galvanometer  of  5000  ohms  resistance,  .0004  of  a second;  2700 
ohm  coil  of  a mirror  galvanometer,  .001  of  a second;  100,000- 
ohm  coil  of  a mirror  galvanometer,  .0007  of  a second;  coil  of 
Ayrton  and  Perry  spring  voltmeter,  .0044  of  a second ; polar- 
ized relays,  types  1,  2,  3,  and  4,  respectively,  .0048,  .0045,  .0041, 
and  .0032  of  a second;  Morse  relays,  from  about  .070  to  .026 
of  a second,  with  about  .034  of  a second  as  an  average  for  in- 
struments in  working  adjustment;  two  telegraph  sounders, 
.0095  and  .0065  of  a second  ; modern  telephone  induction  coil 
primary  about  .006  of  a second,  secondary  of  same  nearly  .003 
of  a second;  a mile  of  bare  No.  12  B.  & S.  gauge  copper  wire 
on  a pole  line,  with  ground  return,  .00037  of  a second;  No.  6 
wire  in  a similar  position,  with  ground  return,  .0014  of  a sec- 
ond; primary  of  large  induction  coil,  .09  of  a second;  sec- 
ondary of  same,  .065  of  a second;  dynamo  fields,  from  about 
.01  to  10  seconds;  direct-current  dynamo  armatures,  from  .005 


* Art.  44. 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  209 


to  5 seconds ; Mordey  36-kilowatt  2000-volt  alternator  arma- 
ture, .017  of  a second;  Ivapp  60-kilowatt  2000-volt  alternator 
armature,  .035  of  a second;  primary  and  secondary  windings 
of  transformers,  from  several  thousandths  of  a second  to  a 
number  of  seconds.  The  Du  Bois  electro-magnet  has  a time 
constant  of  75  seconds.  Finally,  suppose  6 ohms  is  the  resist- 
ance of  the  magnetizing  coil  figuring  in  the  example  of  Art. 
43.  Then  assuming  the  value  of  L to  be  constant,  which  is 
not  exact  when  the  core  is  iron,  the  time  constant  becomes  in 
the  three  cases,  respectively,  .655,  .785,  and  .0026  of  a second. 

62.  Vector  Relations  of  Current  and  Voltage  in  a Circuit  con- 
taining R,  L , and  C ; Complex  Expressions.  — Where  resistance, 
self-inductance,  and  capacity  are  all  in  series  in  a circuit,  the 
effects  of  the  self-inductance  and  capacity,  being  opposite,  tend 
to  neutralize  each  other,  and  the  diagram  takes  the  form  shown 
in  Fig.  132.  Here  OO  is  the  active  voltage  (Ji2  drop),  OB 


Fig.  132.  — Phase  Diagram  of  a Circuit  containing  Resistance,  Self-inductance, 
and  Capacity  in  Series. 


and  OB'  are  voltages  respectively  equal  to  the  self-inductive 
and  capacity  voltages,  and  OA  is  the  impressed  voltage. 
Since  OB'  is  90°  ahead  and  OB  90°  behind  the  active  vol- 
tage, they  are  exactly  in  opposition  and  tend  to  neutralize  each 
other.  If  they  are  of  equal  values,  the  external  effect  is  as  though 
both  were  absent  and  the  impressed  and  active  voltages  are 
then  identical.  In  Fig.  132  the  self-inductive  voltage  is 
greater  than  the  capacity  voltage  and  their  difference  OF  is  the 
reactive  effect  in  the  circuit.  The  external  effect  is  as  though 
a self-inductive  voltage  OF  were  present  without  capacity.  On 
the  other  hand,  if  OB'  should  be  the  greater,  the  difference 
would  show  a capacity  voltage  equal  to  the  difference  between 


210 


ALTERNATING  CURRENTS 


the  two  reactive  voltages.  Figure  133  shows  the  polygon  of 
the  voltages  shown  in  Fig.  132. 


A 


and  Capacity  in  Series. 

The  complex  expression  representing  the  voltages  in  such  a 


circuit  is 

E — Er  +jEt  — jEc , 

or 

E=Er+j{El-Ec') 

and 

E — .27  (cos  6 +j  sin  0), 

where 

E = Vi?,.2  + (E,  — Ecy 

and 

a f - 1 Ei  — Ec 

Er 

If  Ei  > Ec,  the  imaginary  term  of  the  complex  expression  is 
positive  and  the  equation  is  similar  to  one  for  a circuit  contain- 
ing resistance  and  self-inductance  only,  and  the  angle  6 is  posi- 
tive. If  Et  < Ec , the  imaginary  term  is  negative,  and  the  angle 
6 is  negative. 

The  voltage  arising  from  the  combination  of  the  voltages  of 
several  circuits  in  series  is 

E = (En  + Er„  + etc.)  -\-  j(^Eti  -(-  Eln  + etc.  ECi  ECi  etc.) 

■ = S^r+/(2^-2Jre), 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  211 


and  E = E (cos  0 +j  sin  6),  where 

, 'LE,  -IE 

E=  V(^,)2  + (2-Ei  - 2^,)2,  and  6 = tan"1 ' c. 

Either  or  both  E,  and  Ec  may  be  zero  in  any  of  the  partial  cir- 
cuits and  Er  may  be  so  small  as  to  be  negligible.*  The  angle 
6 is  the  angle  of  lag  of  the  circuit.  It  is  measured  from  the 
initial  axis  or  axis  of  reals  (which  gives  the  direction  of  the 
current  vector  or  IR  drop)  towards  the  vector  of  impressed 
voltage.  When  6 is  positive,  the  circuit  carries  a current  which 
lags  with  respect  to  the  voltage  impressed  at  the  circuit  termi- 
nals ; and  when  6 is  negative,  the  circuit  carries  a current 
which  leads  with  respect  to  the  voltage  impressed  at  the 
terminals. 

The  current  in  a circuit  may  be  represented  by  a vector 
(complex)  equation  in  the  same  manner  as  a voltage.  In  this 
case  the  current  must  be  resolved  into  rectangular  components, 
preferably  with  one  component  parallel  to  the  vector  of  voltage 
impressed  on  the  circuit.  It  will  be  shown  later  that  the  initial 
axis  in  this  case  should  correspond  with  the  direction  of  the 
vector  of  voltage  impressed  on  the  circuit,  and  therefore  the 
angle  measured  from  the  initial  axis  to  the  current  vector  is 
the  angle  of  lag  reversed  ; that  is,  it  is  — (±6). 

Knowing  the  rectangular  components  of  the  several  currents 
in  a number  of  branches  or  parallel  circuits,  the  components 
of  the  main  current  are  obtained  by  adding  the  corresponding 
components  of  the  branch  currents  together  algebraically  ; as 

1=  tlr  +/(2/r), 

and  1=  /(  + cos  0 — j sin  #). 

In  this  case 
and 


J=  V(2J,)!  + (S4)2 

V T 

6 = tan-1 


63.  Impedance  and  Reactance  and  their  Expression  as  Com- 
plex Quantities. — The  quantity 


\/*“2 


+ 27 rfL- 


J_Y 


called  the  Impedance  of  the  circuit  and  ( 2 irfL  — 


2 TrfCj 

1 


2 irfC 


is 


is 


* Art.  58. 


212 


ALTERNATING  CURRENTS 


called  tlie  Reactance,  whatever  may  be  the  values  of  R,  L , or  C. 
The  square  of  the  impedance  of  a circuit  is  therefore  equal  to 
the  sum  of  the  squares  of  its  resistance  and  reactance.  Imped- 
ance and  reactance  are  both  of  the  dimensions  of  resistance 
and  are  therefore  expressed  in  ohms.  Impedance  may  be  de- 
fined for  circuits  in  general,  as  the  total  opposition  in  a circuit 
to  the  flow  of  an  alternating  electric  current,  or  the  ratio  of  the 
voltage  to  the  current;  and  Reactance  may  be  defined  as  the 
component  of  the  impedance  caused  by  the  self-inductance  and 
capacity  of  the  circuit. 

The  reciprocal  of  impedance  is  called  Admittance.  It  is 
equal  to  the  current  divided  by  the  voltage.  It  therefore 
measures  the  tendency  of  a voltage  to  force  current  through  a 
circuit.  It  is  measured  in  Mhos  or  the  reciprocal  of  ohms,  as 
the  term  indicates. 

The  Capacity  Reactance  of  a circuit  is  inversely  proportional 
to  the  capacity  and  the  frequency  in  the  circuit,  since  it  is 


equal  to  — 


rrfC' 


The  Inductive  Reactance  of  a circuit  is  directly 


proportional  to  the  self-inductance  and  the  frequency  in  the 
circuit,  since  it  is  equal  to  2 irfL.  The  total  reactance  of  a cir- 
cuit containing  capacity  and  self-inductance  in  series  bears  a 
complicated  relation  to  the  capacity,  self-inductance,  and  fre- 


quency in  the  circuit,  since  it  is  equal  to  2 Itis 

-irfl 


zero  when,  for  any  reason,  2 n rfL  and 


are  both  equal  to 


2 irfC 

zero,  which  is  the  condition  of  a circuit  of  resistance  alone ; or 


whenever  2 7 rfL  = , 

J 2 irfO 


that  is,  when  2nf=  — 

Vic7 


Polygons  of  impedance  may  be  directly  obtained  from  the 
polygons  of  voltages  as  are  shown  in  Figs.  119,  128,  and  133, 
by  dividing  each  side  of  the  polygon  by  the  current  I.  Figure 
134  shows  the  impedance  diagram  for  a self -inductive  circuit; 
Fig.  135  for  a capacity  circuit;  and  Fig.  136  for  a series 
circuit  which  includes  both  self-induction  and  capacity.  The 
capacity  effect  predominates  over  the  self-inductance  in  the  latter 
figure,  which  in  this  differs  from  the  circuit  represented  in 
Fig.  133.  Impedance  is  usually  represented  in  formulas  by  the 
letter  Z,  reactance  by  the  letter  X , and  admittance  by  the  letter  Ti 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  213 


Impedance  may  be  treated  as  a vector  operator ; that  is,  as  a 
line  having  definite  length  and  direction.  It  may  be  expressed 
in  the  complex  form  in  this  way, 

Z=R+jX 

and  Z - Z( cos  0 +j  sin  ; 

also  Z = V R2  + X2  and  0 = tan-1  — • 

R 


Circuit. 


2?r/C 


Fig.  135.  — Impedance  Triangle  for  a Capacity  Circuit. 


ing  both  Self-induction  and  Capacity,  Capacity  Pre- 
dominating. 

Since  the  triangles  of  impedance  are  geometrically  similar  to 
the  corresponding  triangles  of  voltage,  the  angle  0 is  the  same 
in  each,  i.e.,  it  is  the  angle  of  lag  in  the  circuit.  If  X and  0 
are  negative,  capacity  predominates,  and  the  imaginary  terms 
become  — ■ jX  and  — j sin  0.* 


* Art.  81. 


214 


ALTERNATING  CURRENTS 


The  combined  impedance  of  a number  of  circuits  in  series  is 
Z = (R1  4-  -B2  + etc. ) j(^X^  4"  X2  4-  etc. ) = ~R  4-  j^LX, 

in  which  the  reactances  are  to  be  added  algebraically ; 

Z = Z(cos  0 +j  sin  6 ) 


and 


tan  0 - 


IT 

17? 


Admittance  being  the  reciprocal  of  impedance, 


1_  1 _ R —jX  R . X 

Z R+jX  R2  + A"2  R2  4 X2  ^ R?  + X2' 


Putting  g ani  b for  the  horizontal  and  vertical  components, 
respectively,  gives 


Y = 


R . X ., 

t*>  . = 


Thus,  g = 
Also,  b - 


R 


R2  + X2 


R2  + X2  ' R2+X 2 
and  is  called  the  Conductance  of  the  circuit. 


X 


R2  + X2 


and  is  called  the  Susceptance  of  the  circuit. 
g R 

Admittances  in  parallel  can  be  combined  in  the  same  manner 
as  impedances  in  series. 


Then, 


Assume  sinusoidal  voltages  and  currents  in  the  following 
problems : 

Prob.  1.  A circuit  has  a resistance  of  10  ohms  and  a capacity 
of  50  microfarads  in  series.  What  is  the  impedance  of  the  cir- 
cuit when  the  frequency  is  60  periods  per  second?  Solve  by 
the  graphical  method  and  by  means  of  complex  quantities. 

Prob.  2.  A circuit  has  a resistance  of  50  ohms  and  a capacity 
of  200  microfarads  in  series.  What  is  the  impedance  when  the 
frequency  is  25  periods  per  second? 

Prob.  3.  If  the  resistance  of  a capacity  circuit  is  20  ohms 
and  its  impedance  is  50  ohms,  what  is  the  capacity  in  micro- 
farads, and  what  is  the  angle  of  lag,  the  frequency  being  120 
periods  per  second?  Solve  graphically  and  analytically. 

Prob.  4.  A circuit  has  an  impedance  of  100  ohms.  The 
frequency  is  60  periods  per  second  and  the  angle  of  lag  is  45°. 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  215 

Find  the  resistance  and  inductance  of  the  circuit  by  graphics 
and  by  vector  equations. 

Prob.  5.  A circuit  has  an  impedance  of  200  ohms.  The 
angle  of  lag  of  the  circuit  is  —60°  and  the  frequency  is  25 
periods  per  second.  Find  the  resistance  and  capacity  by 
graphics  and  by  vector  equations. 

Prob.  6.  A circuit  having  an  angle  of  lag  of  — 90°  and  a 
frequency  of  40  periods  per  second  has  an  impedance  of  50 
ohms.  What  is  the  resistance  and  what  is  the  capacity?  Solve 
by  graphics  and  by  vector  equations. 

Prob.  7.  A circuit  in  which  there  is  an  angle  of  lag  of  90° 
and  a frequency  of  40  periods  per  second  has  an  impedance  of 
100  ohms.  Find  the  resistance  and  the  reactance  by  graphics 
and  by  vector  equations. 

Prob.  8.  A circuit  has  a resistance  of  10  ohms,  a self-induct- 
ance of  .01  of  a henry,  and  a capacity  of  50  microfarads  in  series. 
If  the  frequency  is  60  periods  per  second,  what  is  the  value  of 
the  impedance  and  of  the  angle  of  lag? 

Prob.  9.  A circuit  has  a resistance  of  20  ohms,  a self-induct- 
ance of  .01  of  a henry,  and  a capacity  of  200  microfarads  in 
series.  Find  the  impedance  and  angle  of  lag  by  graphics  and 
vector  equations  when  the  frequency  is  40  periods  per  second. 

Prob.  10.  A circuit  has  a resistance  of  50  ohms,  a self- 
inductance of  .02  of  a henry,  and  a capacity  of  20  microfarads 
in  series.  Find  the  impedance  and  angle  of  lag  by  graphics  and 
vector  equations  when  the  frequency  is  25  periods  per  second. 

Prob.  11.  A circuit  has  an  inappreciable  resistance,  a self- 
inductance of  .01  of  a henry,  and  a capacity  of  100  microfarads 
in  series.  Find  the  impedance  and  the  angle  of  lag  when  the 
frequency  is  60  periods  per  second. 

Prob.  12.  Two  circuits  having  self -inductances  of  .01  and  .02 
of  a henry  and  resistances  of  8 ohms  and  10  ohms,  respectively, 
are  in  series.  Find  their  combined  impedance  and  the  angle  of 
lag  of  the  circuit  when  the  frequency  is  40  periods  per  second. 

Prob.  13.  Two  circuits  having  capacities  of  100  and  200 
microfarads  and  resistances  of  10  and  20  ohms,  respectively,  are 
in  series.  Find  the  impedance  and  angle  of  lag  when  the  fre- 
quency is  40  periods  per  second. 


216 


ALTERNATING  CURRENTS 


Prob.  14.  Two  circuits  have  resistances  respectively  of  10 
and  15  ohms,  the  first  has  a capacity  of  100  microfarads  and 
the  second  a self-inductance  of  .01  of  a henry.  If  the  circuits 
are  connected  in  series,  what  is  the  combined  impedance  and 
what  is  the  angle  of  lag  when  the  frequency  is  40  periods  per 
second? 

Prob.  15.  If  a voltage  of  100  volts  is  impressed  upon  the 
joint  circuit  of  problem  14,  how  much  current  will  flow? 

Prob.  16.  If  it  is  desired  to  cause  a current  of  10  amperes 
to  flow  through  the  joint  circuit  of  problem  14,  what  voltage 
must  be  impressed? 

Prob.  17.  A circuit  has  a resistance  of  10  ohms,  a self-induct- 
ance of  .01  of  a henry,  and  a capacity  of  200  microfarads  all 
in  series,  and  a voltage  of  200  volts  with  a frequency  of  60 
periods  per  second  is  impressed  on  this  circuit.  What  is  the 
active,  and  what  is  the  reactive,  voltage?  Find  graphically  and 
by  vector  equations  the  current  flowing  through  the  circuit. 

Prob.  18.  Three  coils  of  respectively  1 ohm,  2 ohms,  and 
3 ohms  resistance  having  self-inductances  of  .01,  .02,  and  .03 
of  a henry  are  connected  in  series.  If  a current  of  10  amperes 
flows  through  the  series  at  a frequency  of  25  periods  per  second, 
what  is  the  voltage  between  the  terminals  of  each  coil,  and  what 
is  the  voltage  between  the  terminals  of  the  series  ? 

Prob.  19.  Three  condensers  having  internal  resistances  of 
8,  10,  and  12  ohms,  respectively,  and  capacities  of  50,  100,  and 
180  microfarads  are  connected  in  series.  What  is  the  total 
voltage  impressed  on  the  series  and  what  is  the  voltage  be- 
tween the  terminals  of  each  condenser  when  the  current  is  20 
amperes  with  a frequency  of  60  periods  per  second  ? 

Prob.  20.  A circuit  of  10  ohms  resistance  and  .01  of  a henry 
self-inductance  is  in  series  with  a circuit  of  8 ohms  resistance 
and  200  microfarads  capacity  and  these  in  turn  are  in  series 
with  another  circuit  of  15  ohms  resistance,  .02  of  a henry  self- 
inductance, and  150  microfarads  capacity.  When  a current  of 
5 amperes  with  a frequency  of  40  periods  per  second  is  caused 
to  flow  through  this  series,  what  are  the  voltage  and  angle  of 
lag  observed  at  the  main  terminals,  and  what  are  the  voltages 
and  angles  of  lag  observed  at  the  terminals  of  each  of  the  three 
above-named  parts  of  the  total  circuit? 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  217 


Prob.  21.  The  current  in  a circuit  is  50  amperes  with  a 
period  of  one  one-hundred-and-twentieth  of  a second.  The  cir- 
cuit contains  a positive  (inductive)  reactance  of  20  ohms  and 
a resistance  of  20  ohms.  What  are  the  impressed  voltage  and 
the  angle  of  lag  ? 

Prob.  22.  Twenty  amperes  flow  through  a circuit  with  an 
angle  of  lag  of  30°,  when  the  impressed  voltage  is  100  volts 
and  the  frequency  40  periods  per  second.  What  are  the  imped- 
ance, reactance,  and  resistance  ? 

Prob.  23.  A circuit  having  an  impedance  of  40  ohms  is  in 
series  with  a circuit  having  an  impedance  of  30  ohms.  When 
a voltage  of  100  volts  with  a frequency  of  60  periods  per  second 
is  applied  to  the  two  circuits,  the  angle  of  lag  in  the  first  cir- 
cuit is  30°  and  in  the  second  circuit  is  — 60°.  How  much 
current  flows  through  the  circuits  and  what  is  the  angle  of  lag 
observed  at  the  main  terminals  ? 

Prob.  24.  Three  circuits  are  in  series : the  first  has  a re- 
sistance of  20  ohms,  the  second  a resistance  of  10  ohms  and  a 
positive  (inductive)  reactance  of  20  ohms,  the  third  a resistance 
of  5 ohms  and  a negative  (capacity)  reactance  of  30  ohms,  the 
frequency  being  25  periods  per  second.  What  is  the  angle  of 
lag  observed  at  the  main  terminals  and  what  voltage  is  im- 
pressed between  the  main  terminals  when  a current  of  50 
amperes  flows  through  the  series  ? 

Prob.  25.  What  is  the  angle  of  lag  and  what  is  the  voltage 
observed  between  the  terminals  of  each  of  the  three  parts  of 
the  main  circuit  of  problem  24  ? 

Prob.  26.  If  a voltage  of  500  volts,  frequency  25  periods 
per  second,  is  impressed  on  the  main  circuit  of  problem  24,  what 
current  flows  and  what  is  the  angle  of  lag  ? 

Prob.  27.  What  is  the  voltage  across  each  of  the  three 
parts  of  the  main  circuit  of  problem  24  under  the  conditions  of 
current  flow  fixed  by  problem  26  ? 

Prob.  28.  A circuit  of  negligible  resistance  has  a positive 
(inductive)  reactance  of  10  ohms  when  the  frequency  is  60 
periods  per  second.  What  current  flows  when  a voltage  of 
100  volts  is  impressed  on  the  circuit,  and  what  is  the  angle 
of  lag  ? 


218 


ALTERNATING  CURRENTS 


Prob.  29.  A circuit  of  negligible  resistance  has  a negative 
(capacity)  reactance  of  10  ohms  when  the  frequency  is  60 
periods  per  second.  What  is  the  current  when  the  impressed 
voltage  is  100  volts,  and  what  is  the  angle  of  lag  ? 

Prob.  30.  A circuit  of  negligible  resistance  has  a positive 
reactance  of  10  ohms  and  a negative  reactance  of  10  ohms  when 
the  frequency  is  60  periods  per  second.  What  current  flows 
when  100  volts  are  impressed  on  the  circuit  ? 

Prob.  31.  A circuit  has  a resistance  of  10  ohms,  a positive 
reactance  of  10  ohms,  and  a negative  reactance  of  10  ohms, 
when  the  frequency  is  60  periods  per  second.  What  current 
flows  when  a voltage  of  100  volts  is  impressed  upon  the  circuit, 
and  what  is  the  angle  of  lag  ? 

Prob.  32.  A circuit  of  negligible  reactance  has  a resistance 
of  10  ohms.  When  a voltage  of  100  volts  at  a frequency  of  60 
periods  per  second  is  impressed  upon  the  circuit,  what  current 
flows  and  what  is  the  angle  of  lag? 

Prob.  33.  Find  the  self-inductance  and  capacity  in  each  of 
the  circuits  given  in  problems  28  to  32,  inclusive. 

64.  Application.  — The  application  to  circuits  in  general  when 
the  currents  and  voltages  are  sinusoidal,  and  to  alternator  arma- 
tures in  particular,  of  the  preceding  deductions  is  evident. 

Thus,  suppose  it  is  desired  to  design  an  alternator  which  is 
to  generate  25  amperes  at  an  effective  voltage  of  1000  volts  at 
its  terminals  when  operating  on  a non-reactive  load,  the  fre- 
quency being  100  periods  per  second.  Take  first,  for  example, 
a disk  armature  without  iron  in  its  core,  with  a resistance  of 
1 ohm  and  an  average  self-inductance  of  .01  henry.  The  effec- 
tive value  of  the  total  voltage  to  be  developed  in  this  armature 
at  full  load  is  then 

E = Er  +jEx, 

E = VA,2  + E}  = _ZV  A2  + (2  t r/A)2, 

E = ,V(1000  + 25  x 1)2+  (2tt  x 100  x .01  x 25)2, 

which  is  equal  to  1037  volts.  Consequently,  the  effect  of  self- 
inductance is  to  demand  an  increase  of  the  total  voltage  equal 
to  12  volts.  Suppose,  however,  the  armature  is  of  a type  hav- 
ing an  iron  core  and  has  a resistance  of  1 ohm  and  an  average 


SELF-INDUCTION,  CAPACITY,  KEACTANCE,  AND  IMPEDANCE  219 

working  inductance  of  .05  henry,  the  total  voltage  then  becomes 

E = V(1000  + 25  x l)2  + (2  7T  x 100  x .05  x 25)2, 

which  is  equal  to  1291  volts.  Hence,  the  total  voltage  must  be 
increased  by  266  volts  on  account  of  self-inductance.  If  the 
two  machines  were  worked  at  full  load  upon  resistances  of 
absolutely  no  inductance  or  capacity,  the  lags  of  the  currents 
with  respect  to  the  induced  voltages  in  the  circuits  in  the  two 
cases  would  be  respectively  8°  43'  and  37°  27'  (Fig.  137). 

JE 

These  values  are  obtained  from  the  relation  6 — tan-1-^- 

On  the  other  hand, 
in  case  the  load  con- 
tains reactance,  the  fig- 
ures are  modified.  For 
instance,  in  case  the  load 
contains  40  ohms  of  im- 
pedance as  before,  but 
in  this  instance  is  com- 
posed of  32  ohms  resist- 
ance  and  24  ohms 
positive  (inductive) 
reactance,  the  induced 
voltage  required  to  af- 
ford 1000  volts  at  the  E _IR 

terminals  of  the  genera-  Fig  137._Voltage  Triangle  calculated  for  an 
tor  with  the  iron-cored  Alternator  Armature, 

armature  is 

E = V[25(l  + 32)  ]2  + [25(31.4  + 24)p, 

which  is  equal  to  1612  volts ; while  if  the  reactance  of  the  load 
is  negative  instead  of  positive, 

E=  V[25(l  + 32)]2  + [25(31.4  - 24)]2, 

which  is  equal  to  845  volts.  In  each  case  the  terminal  voltage 
of  the  generator,  that  is,  the  voltage  impressed  on  the  load,  is 
1000  volts ; but  the  voltage  impressed  on  the  entire  circuit,  that 
is,  the  induced  voltage,  is  affected  by  the  relations  of  the  im- 
pedance of  the  load  to  the  impedance  of  the  armature  winding. 
In  case  the  impedance  of  the  load  contains  negative  reactance, 


220 


ALTERNATING  CURRENTS 


it  is  to  be  observed  that  the  terminal  voltage  may  be  actually 
higher  than  the  induced  voltage  on  account  of  the  interaction 
of  the  capacity  of  the  load  on  the  inductance  of  the  armature 
winding. 

Prob.  1.  An  alternator  with  terminal  voltage  of  10,000  volts 
and  frequency  of  60  periods  per  second  furnishes  50  amperes  to 
an  external  circuit.  The  armature  has  a resistance  of  4 ohms 
and  a self-inductance  of  .02  of  a henry.  When  the  external 
load  is  non-reactive,  what  total  voltage  must  be  generated  by 
the  machine  ? 

Note.  — This  and  the  following  problems  are  to  be  solved  by  the  graphi- 
cal method  and  by  the  use  of  the  complex  (vector)  equations,  neglecting 
any  effects  of  armature  reactions. 

Prob.  2.  An  alternator  generates  a total  voltage  of  5000 
volts  at  a frequency  of  40  periods  per  second.  The  armature 
has  2 ohms  resistance  and  .03  of  a henry  inductance.  How 
much  current  will  this  alternator  deliver  to  a non-reactive  load 
of  20  ohms,  and  what  is  the  terminal  voltage?  Also  what  is 
the  value  of  the  angle  of  lag  between  the  current  and  the  ter- 
minal voltage  and  the  angle  of  lag  between  the  current  and  the 
total  generated  voltage? 

Prob.  3.  The  generator  of  problem  2 is  connected  to  a load 
containing  10  ohms  resistance  and  200  microfarads  capacity  in 
series.  What  are  the  values  of  current,  terminal  voltage,  and 
angle  of  lag  between  current  and  terminal  voltage? 

Prob.  4.  The  generator  of  problem  2 is  connected  to  a load 
containing  20  ohms  resistance  and  .02  of  a henry  self-induct- 
ance in  series.  What  are  the  values  of  current,  terminal  vol- 
tage, and  angle  of  lag  between  the  current  and  terminal  voltage  ? 

Prob.  5.  An  alternator  armature  has  a resistance  of  2 ohms 
and  a self-inductance  of  .02  of  a henry.  It  produces  a termi- 
nal voltage  of  2000  volts  at  a frequenc}r  of  100  periods  per 
second.  Twenty-five  amperes  flow  in  the  circuit,  the  angle  of 
lag  between  the  current  and  the  induced  voltage  is  30°.  What 
are  the  impedance,  resistance,  reactance,  and  self-inductance  of 
the  external  circuit? 

Prob.  6.  In  problem  5 the  angle  of  lag  changes  from  30°  to 
— 60°  and  the  current  to  50  amperes  without  changing  termi- 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  221 

nal  voltage  or  frequency.  What  then  are  the  impedance,  resist- 
ance, reactance,  and  capacity  of  the  external  circuit  ? 

Prob.  7.  The  external  circuit  in  problem  5 again  changes  so 
that  the  tangent  of  the  angle  of  lag  is  .4652 ; the  current  80 
amperes.  What  are  the  impedance,  resistance,  and  reactance  of 
the  external  circuit? 

Prob.  8.  What  are  the  values  of  induced  voltage  correspond- 
ing respectively  to  the  conditions  in  problems  5,  6,  and  7 ; and 
what  are  the  respective  angles  of  lag  between  the  currents  and 
the  induced  voltages? 

65.  Equivalent  Resistance  and  Reactance.  — When  there  is  no 
iron  near  an  alternating-current  circuit,  the  self-inductance  is 
constant  and  the  triangles  of  voltage 
given  previously  are  true  for  all  values 
of  the  current,  but  if  iron  is  present,  L 
varies  with  the  current.  In  addition  to 
this,  if  iron  is  present,  hysteresis  and 
eddy  currents  absorb  power.  Under  a 
given  impressed  voltage  this  power 
reduces  the  reactive  voltage  and  in- 
creases the  active  voltage,  so  that  instead 
of  leaving  the  triangle  of  voltages  OAB 
(Fig.  138)  such  as  would  be  obtained  Fig  138.  — Triangles  of  Vol- 
in a certain  circuit  when  a current  I is  tages  with  and  without  Iron 
flowing  and  there  are  no  Iron  losses  Losses' 
present,  the  triangle  actually  becomes  more  like  OCD.  The 
reactive  voltage  is  reduced  and  the  active  voltage  is  greater  in 
OCD  than  in  OAB , and  the  angle  of  lag  decreased.  The  Equiva- 
lent or  Working  reactance  of  the  circuit  is 
-P7  E'x  Esm  6' 

x =T-—I-. 

■ The  Equivalent  or  Working  resistance  is 

E'r  E'cos0' 

~ I I 

These  conditions  are  discussed  further  in  Chapter  IX. 

Prob.  1.  A circuit  has  a voltage  applied  to-  it  of  100  volts 
and  a current  of  10  amperes  flows  with  a lag  of  60°.  What 
is  the  apparent  resistance  and  what  is  the  equivalent  reactance 
of  the  circuit? 


222 


ALTERNATING  CURRENTS 


66.  General  Equation  for  Current  in  a Circuit.  — The  voltage 
impressed  upon  any  circuit  has  already  been  expressed  * in  the 
form  T ,. 

iK+“  + $— W 


in  which  f(f)  represents  any  voltage  which  may  be  impressed 
on  the  circuit.  This  general  solution  therefore  includes,  when 
properly  interpreted,  the  various  cases  already  discussed  in 
Arts.  45,  52,  and  58. 


Since  ^ idt  = q , 


(1)  may  be  written 


III  di 
~L+dt 


e 

L' 


(2) 


Differentiating  this  with  respect  to  time  to  rid  the  equation  of 
the  integral  sign,  there  results  : 

dI) 2i  Rdi  i _ 1 de  _ 1 
di2  + LJtLC~lTt~L 

The  term  — can  only  have  one  finite  value  in  an  electric  cir- 
dt  J 

cuit  at  any  specific  time,  and  is  therefore  a single-valued  function 
of  time,  which  for  convenience  is  called /'(t).  Equation  (3) 
is  a linear  differential  equation  of  the  second  order  with  its 
second  member  a function  of  t.f 

The  object  is  to  find  the  value  of  current  i,  in  terms  of  R,  L, 
C,  and  the  voltage. 

The  equation  may  be  written 


/'(*)•  (3) 


RD  IV  /'(Q 
L + LG)  L ’ 


(■*) 


where  D is  a symbol  of  operation  and  represents 


*1 1 

dt 


and 


I)2  means  differentiation  twice  with  respect  to  t.  I)  may  be 
used  as  an  ordinary  algebraic  term.J 

j?D  1 

The  “auxiliary  equation”  D2  + -j-  + ^~^,=  0 is  a quadratic 
equation  with  the  two  roots, 


* Art.  58. 

t Price’s  Calculus,  Vol.  II,  p.  458;  Forsyth’s  Differential  Equations,  p.  86; 
Perry’s  Calculus  for  Engineers,  pp.  213-241 ; John  Graham,  An  Elementary 
Treatise  on  Calculus,  p.  221. 

1 Perry’s  Calculus  for  Engineers,  p.  231;  Forsyth’s  Differential  Equations, 
p.  43 ; Murray’s  Differential  Equations,  Chap.  VI. 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  223 


and 


A + , 

2 L 

IB? 

1 

\R  i 

J1R2 

1 -1 

^4  U 

LC 

_2  L ^ 

MX2 

R y 

\IB? 

1 

\h+' 

J1  B? 

1 "| 

2 L ^ 

Mi2 

LC~ 

M X2 

(5) 

(6) 


Letting  ax  and  a2  equal  the  bracket  expressions  respectively 
of  the  second  members  of  equations  (5)  and  (6),  then  the 
“ complementary  function  ” is 

Ae~aJ  + Be~a?.  (7) 

The  particular  integral  is 

1 


zf(t) 


-/'(  0 


7)2  I BI>  i JL  C-®  + (D  + «2) 

L LC 


- z/,(f) 


;} 


(8) 


_I)  T rtj  D -f-  <x2 . 

in  which  the  third  member  is  obtained  by  resolving  the  second 
member  into  partial  fractions.  This  solves  into  the  form, 

1 

L 


Nxe  + (t)dt 


(9) 


The  values  of  Nx  and  iV2  may  be  determined  from  the  identi- 
cal equation,* 


= ^_  + 


2V„ 


whence 


and 


(D  -f-  a1)(i>  -)-  (1%)  B q-  $q  B -(-  ct2 

+ «2)  + N2(^D  + «j)  = 1, 

B(Nt  + N2)  = 0,  Nxa2  + N2a1  = 1, 

jV,=  -V2  = - 1 1 


■i  2JIAJ_A 

'4  i2  LC 


(10) 

(11) 

(12) 

(13) 


The  complete  solution  of  equation  (3),  being  the  sum  of  the 
complementary  function  and  the  particular  integral,  is 

1 


* = 


2 iJT®_  J_ 

V 1 T 2 TO 


— |^e  — e ea~l  f (t)dt 


4 1?  LC 

+ Ae~°lt  + Be _“2<. 

* Wentworth’s  Complete  Algebra , p.  409. 


(14) 


224 


ALTERNATING  CURRENTS 


The  coefficient  of  the  first  term  of  the  second  member  of  the 
last  equation  may  also  be  written 

C 

V_Z22C'2  — 4 Lc' 

The  terms  a1  and  a2  may  also  be  written 

R \Tr?  T _RC-  CICC2  -4  L C 

2 L '4  L2  LC  2 LG 

R ilR 2 T~  RC  + VR?C2  — 4 LC 

2L  + ^4  L2  LC  2 LC 

A similar  equation  can  be  obtained  in  the  same  way  for  the 
charge  at  any  instant  in  a condenser  in  the  circuit. 

67.  Current  in  a Circuit  when  Any  Periodic  Voltage  is  Im- 
pressed.— In  the  case  of  a circuit  with  resistance,  self-induct- 
ance, and  capacity  in  series  on  which  any  periodic  voltage  is 
impressed,  the  voltage  formula  is 

e =/(f)  = iR  4-  L — + — C idt. 

at  C*7  o 

The  general  solution  of  this  is  equation  (14),  Art.  66. 

This  contains  too  many  unknown  terms  to  be  of  much  use  to 
the  electrical  engineer  at  present;  but  if  e=f(t')  is  construed 
to  mean  an  alternating  voltage,  it  may  be  expressed  in  terms  of 
the  harmonics.  This  formula  becomes 


e = /( 0 = emx  sin  («  + ^:)  + e„,3  sin  (3  a + #3)  + ••• 

+ eTOjsin  (no.  + 5>n), 

and 

j~t  =/( 0 = "e,nicos  («  + 0j)  + 3 cos  (3  « + 0g)  ••• 

+ ncoem  cos  ( na  + ^„). 

Substituting  this  value  of  f(t)  in  the  general  equation  for 
current  developed  in  Art.  66,  there  results  an  equation  in  which 
there  is  a pair  of  similar  terms  for  each  harmonic.  These  may 
evidently  be  integrated  by  parts  and  the  result  is  as  follows : 


sin  (a  + tfj) 


i = 


SELF-INDUCTION,  CAPACITY,  KEACTANCE,  AND  IMPEDANCE  225 


em 

H ■■  ■■  3 — sin  (3  u + do) 

+ 

H n - sin  (na  + 0 ) 

+ Ae~aii  + Be~a*. 

The  angle  0n  is  determined  by  the  relation 


tan  0n 


2 t rf  nL  1 = Xn 

I t 2 irfnCR  II ' 


Each  of  the  terms  of  the  right-hand  member,  from  the  first  to 
the  wth,  has  the  form  of  a sine  function. 

The  formula  may  be  more  conveniently  written 

i = — 1 sin  («  + 6X)  + — 'sin  (3  a + 03)  ... 

zx  z3 

+ >sin  (na  + 0n), 

where  Zv  Z3,  •••  ZH  each  represents  the  impedance  which  the 
circuit  offers  to  the  corresponding  current  harmonic.  The 
exponential  terms  are  omitted,  as  they  ordinarily  disappear 
quickly  after  the  current  is  started  ; and  the  formula  repre- 
sents the  conditions  in  the  circuit  after  a permanent  state  is- 
established.  It  will  be  noted  that  the  impedances  for  the  several 
harmonics  may  have  very  different  values  even  though  they  are 
all  affected  by  the  same  resistance,  self-inductance,  and  capacity. 
This  is  due  to  the  frequency  of  the  harmonic  entering  into  the 
reactance  component  of  the  impedance,  as  shown  by  the  formula. 
Each  term  of  the  second  member  represents  the  instantaneous 

value  at  the  instant  t,(^  = — of  a sine  wave  current.  A similar 

formula  may  obviously  be  derived  to  represent  the  condition 
when  even-numbered  harmonics  as  well  as  odd-numbered  har- 
monics appear  in  the  voltage  wave. 

The  effective  value  of  current  is  found  by  taking  the  square 
root  of  the  sum  of  the  squares  of  the  effective  values  of  the 


226 


ALTERNATING  CURRENTS 


harmonic  currents  given  in  the  last  expression,  or 

i=y/r,  + r,+  -p,* 

^vfVtc. 

Prob.  1.  A certain  voltage  wave  is  composed  of  two  har- 
monics, the  fundamental  having  an  effective  value  of  150  volts 
and  that  of  three  times  the  frequency  having  an  effective  value 
of  25  volts.  What  is  the  effective  value  of  the  resultant  voltage? 

Prob.  2.  The  impedance  offered  by  a circuit  to  the  primary 
harmonic  of  a certain  voltage  is  20  ohms,  the  effective  value  of 
the  harmonic  being  100  volts ; the  remaining  harmonic  of  the 
voltage  is  of  five  times  the  primary  frequency,  has  an  effective 
value  of  10  volts,  and  overcomes  an  impedance  of  30  ohms. 
Find  the  effective  values  of  the  voltage  and  current. 

Prob.  3.  A current  is  composed  of  two  harmonics,  a primary 
and  one  of  three  times  the  frequency.  The  first  has  an  effective 
value  of  50  amperes  and  flows  through  an  impedance  of  50  ohms. 
The  other  has  an  effective  value  of  20  amperes  and  flows  through 
an  impedance  of  40  ohms.  What  is  the  effective  value  of  the 
voltage  impressed  on  the  circuit  ? 

68.  Irregular  Current  and  Voltage  Waves  expressed  as  Com- 
plex Quantities. — A Fourier’s  series  may  be  used  to  represent 
irregular  current  and  voltage  by  means  of  component  sine  waves, 
as  pointed  out  in  Art.  67.  The  effective  value  of  such  a vol- 
tage is  found  from  the  relation, 

E2  = E\  + E\  + etc.  + E\  + E2X3  + etc.,* 

and  E = V. E\  + E\  + etc.  + E\  + E2Xs  + etc., 

where  Er  and  Ex  are  active  and  reactive  effective  voltages  for 
the  various  harmonics.  This  may  be  written : 

E = V(^2ri  + E\~)  + (. E\  + E2X3)  + etc. 

and  the  term  within  each  bracket  is  evidently  the  square  of 
the  scalar  value  of  the  corresponding  sine  voltage.  A non- 
sinusoidal  alternating  curve  may  be  empirically  indicated  by 
the  expression 

(-£>,,  Era,  etc.)  +j(EXi,  EX2,  EXs,  etc.); 


* Art.  16. 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  227 


which  is  intended  to  convey  the  idea  that  each  harmonic  of  the 
voltage  curve  acts  individually  to  set  up  a current  of  the  same 
frequency  in  the  circuit,  as  though  the  other  harmonics  were 
not  present.  The  square  root  of  the  sum  of  the  squares  of  the 
effective  values  of  the  harmonic  voltages,  as  already  pointed  out, 
gives  the  effective  value  of  the  total  voltage  impressed  on  the 
circuit.  In  the  same  way,  the  square  root  of  the  sum  of  the 
squares  of  the  effective  values  of  the  individual  current  har- 
monics caused  to  flow  in  the  circuit  by  the  harmonics  of  voltage 
gives  the  effective  value  of  the  actual  current  flowing. 

The  irregular  current  relations  are  similar  to  those  given  for 
voltage. 

Triangles  of  voltages  for  each  harmonic  may  be  drawn,  and 
if  the  corresponding  harmonic  of  current  is  divided  into  the 
value  of  each  side,  an  impedance  triangle  results.  Thus, 

represents  the  impedance  offered  to  the  flow  of  the  harmonic 
of  primary  frequency  ; 


= R + jX3 


represents  the  impedance  offered  to  the  flow  of  the  harmonic 
of  three  times  the  frequency,  etc. 

In  obtaining  the  reactances  for  the  different  harmonics,  care 
must  be  taken  to  use  the  proper  frequency  in  the  formula, 


X=2t rfL,  or  X = 


1 

2 7 rfO' 


Thus,  if  the  resistance  in  a circuit  is  10  ohms  and  the  reactance 
for  the  primary  harmonic  is  5,  there  results 

Zj  = 10  +j  5 ; Z3  = 10  +/15  ; Z5  = 10  +j  25,  etc., 

when  the  voltage  and  current  contain  harmonics  of  odd  fre- 
quencies and  the  reactance  is  inductive.  If  the  reactance 
results  from  capacity,  the  expression  is 


Z1  = 10— /5;  Z3=  10— if;  Z5  = 10 -j  f , etc. 

It  will  be  noticed  that  the  resistance  is  the  same  for  all 
harmonics. 


228 


ALTERNATING  CURRENTS 


If  the  empirical  expression  for  current 

C^i’  Ig^  etc.)  + j (1 j,i . Jf,3 , etc.), 


where  and  _ZJ,  are  the  rectangular  components  of  current,  has 
its  several  harmonic  values  divided  by  the  respective  harmonic 
values  of  the  voltage,  the  admittances  which  the  circuit  offers 
to  each  current  harmonic  may  be  derived.  Thus 


where  g and  b represent  the  rectangular  components  of  the 
admittances.  r r 

is  the  admittance  of  the  circuit  for  the  current  harmonic  of 
primary  frequency, 


is  the  admittance  for  the  current  harmonic  of  three  times  the 
primary  frequency,  etc. 

Vector  voltage  may  obviously  be  multiplied  by  admittance 
or  divided  by  impedance  to  give  vector  current;  or  vector  cur- 
rent may  be  divided  by  admittance  or  multiplied  by  impedance 
to  give  vector  voltage.*  This  process  gives  the  relations  of 
the  equivalent  sinusoids  representing  irregular  curves  of  voltage 
and  current.  The  total  vector  impedance  and  admittance  may 
be  written  empirically 

Z i -)-  Z<£  T Zq  -f-  etc. 

= (Ri  + R<i  + Rg  + etc.)  4 - j (A  j + A2  + Ag  + etc.) 

T)  + Vij  + 1 3 + etc. 

= (9i  +92+ 9s  + etc.)  -3  ibi  + h + b3  + etc-) 

[(i^  + R2  + R3  + etc.)2  + (aq  + x2  + x3  + etc.)2]*, 
and  [(#!  + 92+93  + etc.)2  + (bi  + b2  + bs  + etc.)2] -. 

The  process  of  solving  problems  involving  irregular  curves 
where  the  harmonic  components  are  known  is  similar  to  those 


— (,9v  92'  etc.)  -f  f'3,  etc.), 


* Art.  63. 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  229 


already  disclosed  for  sinusoidal  voltages  or  currents,  but  each 
sinusoidal  component  of  voltage  or  current  must  be  dealt  with 
as  if  acting  alone  in  the  circuit,  and  then  the  results  may  be 
combined  to  obtain  the  total  values  according  to  the  theorem 
that  the  effective  value  of  any  single-valued  periodic  curve  is 
equal  to  the  square  root  of  the  sum  of  the  squares  of  the  effec- 
tive values  of  its  harmonics.  It  should  be  noted  that  the 
impedance  obtained  by  dividing  the  total  voltage  by  the  total 
current  is  dependent  upon  the  shapes  of  the  curves  of  voltage  and 
current  on  account  of  the  fact  that  fixed  values  of  self-induct- 
ance and  capacity  do  not  have  the  same  effect  upon  harmonics 
of  different  frequencies. 

Prob.  1.  The  voltage  impressed  on  a circuit  is  composed  of 
two  harmonics.  The  fundamental  has  an  active  component  of 
190  volts  and  a reactive  component  of  —50  volts,  while  the 
other  harmonic  has  three  times  the  frequency,  an  active  com- 
ponent of  25  volts,  and  a reactive  component  of  20  volts. 
What  is  the  value  of  the  voltage  ? 

Prob.  2.  A circuit  contains  a resistance  of  10  ohms  and  self- 
inductance of  .01  henry.  A current  with  a frequency  of  60 
periods  per  second  flows  in  this  circuit.  It  is  composed  of  a 
fundamental  harmonic  of  100  amperes  and  a third  harmonic  of  20 
amperes.  What  voltage  is  impressed  upon  the  circuit  ? (Aid  : 
Multiply  each  harmonic  current  by  the  corresponding  complex 
expression  for  impedance  to  obtain  the  voltage  harmonics.) 

Prob.  3.  What  impedance  is  offered  by  a circuit  in  which 
voltage  J5r  = (1001,  103)  + y(501,  153)  is  impressed  and  current 
I — (50x,  5g)  +/(40x,  103)  flows?  (Aid:  First  find  effective 
voltage  and  effective  current.) 


69.  Variation  in  the  Impedance  offered  to  Current  and  Vol- 
tage Harmonics  of  Different  Frequencies. — Following  out  the 
remarks  of  the  last  paragraph  of  the  preceding  article,  it  is  to 
be  observed  that,  since  the  reactance  of  a reactive  circuit  is 
dependent  on  the  frequency  of  the  current  flowing,  it  is  obvious 
that  the  impedance  interposed  by  a circuit  to  the  different 
harmonics  of  a current  may  differ  for  the  several  harmonics. 
Inductive  reactance  (2irfL')  varies  directly  with  frequency; 


capacity  reactance 


varies  inversely  with  frequency  ; and 


230 


ALTERNATING  CURRENTS 


the  reactance  of  a circuit  containing  both  inductance  and  capac- 
ity ( 2 7 rfL ~~-—— 

J V 2 t rfC 

and  may  increase  or  decrease  as  frequency  increases,  depending 
upon  the  relations  of  the  inductive  and  capacity  parts  of  the 
reactance. 

Under  these  circumstances,  when  an  irregular  alternating 
voltage  is  impressed  on  a circuit,  the  harmonics  of  the  current 
which  flow  will  not  bear  a uniform  ratio  to  the  voltage  har- 
monies  unless  the  circuit  is  non-reactive.  Assuming  the  self- 
inductance and  capacity  to  be  uniform, 

I1  — ^ + I*  + •••  + 

in  which  Ji  = §2’  ^ = etc- 

But  in  a series  circuit 

z*-*+(s^z-s£J)\ 

Zl=B>+(2„AL-^)\ 


^ is  a more  complex  function  of  frequency 


Zl  = & + (2*f„L- 


in  which /p/g,  etc ,,fn  represent  the  frequencies  of  the  several 
harmonics,  the  numerical  value  of  the  subscript  showing  the 
number  of  times  to  multiply  the  fundamental  frequency  to 
arrive  at  the  particular  frequency  under  consideration. 

Scrutinizing  these  equations  brings  out  a number  of  impor- 
tant facts. 

1.  Only  when  reactance  is  negligible  or  the  voltage  sinus- 
oidal can  the  current  flowing  in  a circuit  have  the  same  wave 
form  as  the  impressed  voltage  which  sets  it  up. 

2.  If  the  capacity  reactance  is  negligible,  as  it  ordinarily  is 
in  the  instance  of  a closed  metallic  circuit  of  relatively  short 
length,  inductive  reactance  only  is  to  be  considered.  In  this 
instance  it  will  be  observed  that  the  reactance  opposed  to  each 
current  harmonic  is  directly  proportional  to  the  frequency  of 
the  harmonic  and  the  impedance  is  larger  for  the  harmonics  of 
higher  frequency  than  for  those  of  lower  frequency.  Under 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  231 

these  circumstances,  the  current  produced  by  an  irregular  im- 
pressed voltage  will  be  less  irregular  than  the  voltage  produc- 
ing it ; that  is,  the  inductive  reactance  has  the  effect  of  reducing 
the  amplitude  of  the  higher  harmonics,  and  thus  reducing  the 
ripples  or  irregularities  on  the  current  curve.  The  lag  angles, 
8V  6V  etc.,  of  the  several  current  harmonics  with  respect  to  the 
corresponding  voltage  harmonics  also  differ  from  each  other 
since  R is  unaffected  by  frequency,  which  additionally  alters 
the  form  of  the  current  wave  compared  with  the  voltage  wave. 
The  tangent  of  the  angle  of  lag  between  each  harmonic  of 
current  and  its  corresponding  harmonic  of  voltage  is  obviously 
proportional  to  the  frequency  of  the  particular  harmonic  cur- 
rent, in  this  instance. 

The  above  conditions  are  illustrated  in  Fig.  139.  The  upper 
curves  show,  in  a full  line  the  voltage  wave  E,  and  in  broken 
lines  its  component  harmonic  waves,  which  are  the  fundamental, 
the  third,  and  the  fifth  harmonics.  This  voltage  wave  is  im- 
pressed on  a circuit  containing  resistance  and  inductive  react- 
ance; namely,  10  ohms  resistance  and  ,01  henry  self-inductance, 
the  fundamental  frequency  being  60  cycles  per  second.  The 
lower  curves  in  Fig.  139  indicate  in  broken  lines  the  harmonics 
of  current  corresponding  to  the  harmonics  of  voltage,  while  the 
resultant  current  is  shown  by  a full  line.  The  impedance  in- 
creases with  the  frequency,  that  encountered  by  the  first  har- 
monic being  10.7  ohms,  by  the  third  15.1  ohms,  and  by  the 
fifth  21.4  ohms.  It  will  also  be  observed  that  the  resultant 
wave  of  current  is  less  irregular  than  the  voltage  wave  produc- 
ing it,  showing  the  effect  of  inductance  in  a circuit  in  smooth- 
ing out  the  irregularities  in  the  current  waves  and  dampening 
the  effects  of  the  higher  harmonics,  provided  the  inductance  is 
fixed  in  value. 

3.  If  the  inductive  reactance  is  negligible,  capacity  reac- 
tance only  is  to  be  considered.  In  this  instance  the  reactance 
encountered  by  each  harmonic  of  current  is  inversely  propor- 
tional to  the  frequency  of  the  harmonic,  and  the  impedance 
encountered  by  each  harmonic  is  therefore  smaller  as  the 
frequency  of  the  harmonic  is  larger.  Consequently,  the  cur- 
rent produced  in  a condenser  by  an  irregular  impressed  voltage 
is  more  irregular  than  the  impressed  voltage.  Each  harmonic 
of  the  voltage  is  represented  by  an  harmonic  of  current,  and 


282 


ALTERNATING  CURRENTS 


the  higher  harmonics  are  exaggerated  in  comparison  with  the 
fundamental.  In  the  instance  of  a condenser  without  resist- 


ance, the  relative  amplitudes  of  the  higher  harmonics  of  current 
increase  over  the  relative  amplitudes  of  the  harmonics  of  im- 
pressed voltage  in  direct  proportion  to  their  several  frequencies. 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  233 

The  lag  angles,  6V  d2,  etc.,  of  the  several  current  harmonics 
with  respect  to  the  corresponding  voltage  harmonics,  are  obvi- 
ously smaller  as  the  frequencies  of  the  harmonics  are  greater. 
The  tangent  of  each  angle  of  lag  is  inversely  proportional  to 
the  frequency  of  the  particular  harmonic. 

These  conditions  of  a circuit  containing  capacity  reactance 
are  illustrated  in  Fig.  140.  In  this  case  the  same  voltage  as 


that  shown  in  Fig.  139  is  impressed  on  a circuit  containing  a 
resistance  of  10  ohms  and  a capacity  of  100  microfarads.  The 
harmonics  of  current  corresponding  to  the  harmonics  of  voltage 
are  indicated  in  Fig.  140  by  broken  lines,  and  the  resultant  cur- 
rent is  shown  by  a full  line.  The  impedance  encountered  by 
the  first  harmonic  is  28.4  ohms,  by  the  third  13.4  ohms,  and 
by  the  fifth  11.3  ohms,  while  the  respective  angles  of  lag  for  the 
current  harmonics  are  9X  = 69°  22'  ; d3  = 41°  30' ; = 27°  57'. 


234 


ALTERNATING  CURRENTS 


It  is  observed  that  the  resultant  current  is  more  irregular  than 
the  voltage  producing  it,  showing  that  the  effect  of  capacity  in 
a circuit  is  to  exaggerate  the  higher  harmonics  and  distort  the 
resultant  wave. 

4.  When  the  circuit  contains  both  capacity  and  self-induct- 
ance, these  two  factors  of  the  reactance  vary  differently  with 
frequency  and  the  effects  on  the  current  harmonics  are  not  so 


represent  the  reactance,  it  will  be  observed : that  very  low  fre- 
quencies, in  general,  make  the  first  term  negligible  compared  with 
the  second;  that  very  high  frequencies  make  the  second  term  neg- 
ligible compared  with  the  first  ; and  that  the  two  terms  are 
equal  to  each  other  and  the  reactance  reduces  to  zero  at  some 
intermediate  frequency.  Whatever  may  be  the  particular  nu- 
merical values  of  L and  C in  any  instance,  some  frequency  will 

make  2 7 rfL  = - — — • This  frequency  is,  plainly,  f — ^ 

2 TrfC  1 * >'J  2-kVLC 


A higher  frequency  makes  2 irfL  > 


2-rrfC 


, and  a lower  fre- 


quency makes  2tt/A<- — — — • The  condition  in  which  2 irfL 

1 TTJ  C 

= in  which  X — 0 and  Z — R,  is  called  the  condition  of 

2 TTJ  C 

Resonance,  in  analogy  with  the  tuning  of  a wire  or  a windpipe 
by  adjusting  the  inertia  and  elasticity  of  the  vibrating  medium 
so  as  to  afford  a maximum  natural  amplitude  of  vibration  at  a 
particular  frequency  of  disturbance. 

Applying  the  above  considerations  to  the  circuit  conditions 
wheti,  for  instance,  the  numerical  values  of  L and  C make 

2 7 rfL  < o y-y  for  the  fundamental  frequency  of  the  impressed 

voltage,  it  is  to  be  readily  seen  that  the  impedance  encountered 
by  higher  harmonics  of  current  will  decrease  up  to  a harmonic  of 

the  frequency  at  which  2 irfL  = anc^  the  impedance  over- 


come by  the  harmonic  of  that  particular  frequency  is  equal  to 
the  resistance  of  the  circuit.  Current  harmonics  of  still  higher 
frequency  encounter  impedance  which  is  again  greater  than  R 
and  increases  with  the  frequency.  The  angles  of  lagf  for  the 


236 


ALTERNATING  CURRENTS 


various  harmonics,  as  the  frequencies  increase,  change  from 
negative  values  through  zero  to  positive  values.  In  conse- 
quence of  these  relations,  the  form  of  the  current  wave  in  a cir- 
cuit of  this  character  may  be  greatly  distorted  from  the  form  of 
the  wave  of  impressed  voltage,  through  the  exaggeration  of  a 
certain  harmonic  or  harmonics  and  the  shifting  of  the  relative 
positions  of  the  higher  harmonics  of  current  with  respect  to 
corresponding  voltage  harmonics.  These  conditions  are  illus- 
trated in  Fig.  141.  The  upper  curves  show  a voltage  wave 
in  a full  line  and  its  harmonics  in  broken  lines.  It  is  observed 
that  the  harmonics  are  the  first,  third,  fifth,  seventh,  and  ninth. 
This  voltage  wave  is  impressed  on  a circuit  of  .1  ohm  resistance, 
.3  henry  self-inductance  and  1 microfarad  capacity  in  series. 
The  fundamental  frequency  is  60  periods  per  second.  The 
impedance  encountered  by  the  fundamental  (first)  harmonic  of 
current  is  2539.4  ohms;  the  impedance  encountered  by  the 
third  harmonic  is  544.9  ohms  ; by  the  fifth  harmonic  is  35  ohms, 
as  at  this  frequency  the  condition  of  resonance  is  almost  at- 
tained; the  impedance  encountered  the  seventh  harmonic  is 
412.8  ohms;  and  the  impedance  encountered  by  the  ninth  har- 
monic of  current  is  725  ohms.  The  respective  angles  of  lag 
are  each  nearly  90°,  being  negative  for  the  first  two  harmonics 
and  positive  for  the  others.  The  remarkable  exaggeration  of 
the  fifth  current  harmonic  caused  by  resonance  illustrates 
clearly  the  circumstances  here  discussed. 

Looking  upon  the  foregoing  considerations,  it  is  to  be 
observed  that  the  form  and  magnitude  of  the  voltage  wave 
have  been  assumed,  and  the  conditions  of  current  flow  have  been 
derived.  Converse^,  in  case  a current  of  particular  form  and 
magnitude  is  imposed  on  a circuit,  the  form  and  amplitude  of 
the  voltage  observed  at  the  terminals  of  the  circuit  or  any 
part  thereof  are  dependent  upon  the  current  harmonics  and 
the  constants  of  the  circuit.  In  these  cases  it  is  entirely  pos- 
sible to  have  high,  and  possibly  injurious,  voltages  set  up  in 
parts  of  the  circuit,  although  the  effective  value  of  the  voltage 
at  the  main  circuit  terminals  is  within  ordinary  bounds. 

Prob.  1.  A certain  circuit  has  a self-inductance  of  .01  of  a 
henry  and  a resistance  of  10  ohms,  the  voltage  impressed  upon 
this  circuit  has  a frequency  of  60  periods  per  second.  The 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  237 


voltage  is  composed  of  the  first  four  harmonics  having  the 
respective  effective  values  of  100,  75,  50,  and  25  volts.  What 
current  flows  under  influence  of  each  one  of  these  harmonics,  and 
what  is  the  impedance  opposed  to  the  current  of  each  harmonic  5 
What  is  the  effective  value  of  the  resultant  current  ? 

Prob-  2.  The  voltage  wave  of  problem  1 is  impressed  upon  a 
circuit  of  10  ohms  resistance  and  200  ^microfarads  capacity. 
Answer  the  questions  asked  in  problem  1. 

Prob.  3.  The  voltage  wave  of  problem  1 is  impressed  on  a 
circuit  of  .02  henry  self-inductance  and  50  microfarads  capac- 
ity. Answer  the  questions  asked  in  problem  1. 

Prob.  4.  Answer  problems  1,  2,  and  3 when  the  frequency 
is  changed  to  120  cycles  per  second  without  changing  the  shape 
of  the  voltage  wave. 

Prob.  5.  Answer  problems  1,  2,  and  3 when  the  frequency  is 
changed  to  25  cycles  per  second  without  changing  the  shape  of 
the  voltage  wave. 


Fig.  142.  — Current  Locus  in  Series  Circuit  with  Constant 
Reactance  and  Varied  Resistance. 


70.  The  Envel- 
ope of  the  Current 
Vector  when  the 
Conditions  in  Circuit 
Vary.  — (a)  A con- 
sideration of  the 
effects  of  the  varia- 
tion of  the  con- 
stants in  a series 
circuit  gives  rise  to 
the  following  theorems  for  the  locus  of  the  current  vector 
when  the  voltage  and  frequency  are  fixed. 

Case  (1).  Reactance  Constant  and  Resistance  Varied.  — In 
Fig.  142  let  OR  represent  the  impressed  voltage  R and  OC 

JE 

represent  the  current  J0  = — when  R = 0,  in  which  case  it  is 

X 

obvious  that  OC  must  lag  (or  lead)  OR  by  90°.  Let  OA  rep- 
resent the  current  I for  some  value  of  R between  zero  and 
infinity. 

Now  r=|and  J0  = |- 


238 


ALTERNATING  CURRENTS 


Hence, 

But 


IZ  = I0X  and  I = I0 


X 

Z 


X 

Z 


OB 

OA 


OA 

00 


- cos  6. 


Hence, 


I = I0  cos  6, 


E 

which  is  the  equation  of  a circle  whose  diameter  is  IC)  — — , and 

X 

the  circle  is  the  locus  of  the  current  vector  when  the  reactance 
is  constant  and  the  resistance  varies. 

A consideration  of  the  limits  in  the  variation  of  R shows, 


as  above  stated,  that  when  R = 0,  I— 


E_ 

X' 


and  when  R = oo  , 


1=  0;  and  thus  when  R is  positive,  the  current  vector  is 
limited  to  the  semicircle  OAC.  Now  suppose  a negative  resist- 
ance is  introduced  in  the  circuit,  — R varying  from  0 to  go  . 
Negative  resistance  may  he  physically  considered  as  the  ratio 
of  a voltage  to  the  current  flowing,  where  the  voltage  may  be 
that  of  a generator,  transformer  or  other  means  by  which  me- 
chanical, chemical  or  other  energy  is  transformed  into  electrical 
energy,  instead  of  electrical  energy  being  transformed  into  heat 
or  mechanical  energy  as  in  ordinary  resistance.  The  current 
must  have  a component  in  opposition  to  the  circuit  fall  of 
potential  and  thus  lies  in  the  lower  semicircle  ODC. 


E 

It  will  he  observed  that  the  diameter  OC=  I0  = — will  fall 

A 

c'  90°  to  the  right  or 


C" 

Fig.  143.  — Current  Locus  in  Series  Circuit  with  Constant 
Resistance  and  Varied  Reactance. 


left  of  the  voltage 
vector  OE  according 
as  inductive  react- 
ance or  capacity  re- 
actance p r e d o m i- 
nates.  Thus  in  Fig. 
142  the  circle  to  the 
right  of  the  voltage 
vector  represents 
conditions  when  in- 
ductive reactance 


predominates,  while  the  circle  to  the  left  is  the  locus  of  the 
current  when  capacity  reactance  prevails. 


SELF-INDUCTION,  CAPACITY,  REACTANCE,  AND  IMPEDANCE  239 


Case  (2).  Resistance  Constant  and  Reactance  Varied. — As 
before,  let  OR , Fig.  143,  represent  the  impressed  voltage  and  let 

E 

OC  represent  the  current  I0  = — when  the  reactance  X = 0,  in 

R 

which  case  it  will  obviously  be  in  phase  with  OR  since  there 
is  only  pure  resistance  in  the  circuit.  Let  OA'  represent  the 
current  for  some  value  of  X between  0 and  infinity.  Evi- 
dently when  „ 


Now 

Hence, 

But 

Hence, 


X=cx>  ,1=  0. 

T R ij  R 

I=Z*niI°=R 

1Z  = I0R  and  1=1  A 
Zj 


R 

Z 


OB'  OA' 


OA'  OC 
1=  I0  cos  6', 


= cos  O'. 


Following  the  reasoning  of  Case  (1),  when  R is  considered 
as  positive  and  X is  positive,  the  semicircle  OA' C is  the  limit 
of  the  current  vector,  and  when  X is  negative  the  semicircle 
OA"  C is  the  limit  of  the  vector.  When  R is  considered  as 
negative,  the  current  becomes  opposed  to  the  voltage  and  com- 
prises a component  180°  from  it,  whence  the  circle  OB'  C" I)" 
is  the  limit  of  the  current  vector.  As  in  Case  (1)  the  current 
vector  will  fall  to  the  right  or  left  of  the  voltage  OR  by  an 
angle  O'  of  lag  or  lead,  according  as  inductive  or  capacity 
reactance  predominates. 

( b ) The  converse  of  the  above  cases,  namely,  when  the  resist- 
ance and  reactance  are  constant  but  the  voltage  and  frequency 
vary  in  turn,  indicates  the  following : 

Case  (3).  Voltage  fi-xed  and  Rrequency  Varied.  — Any  varia- 
tion in  the  frequenc}r  must  cause  a variation  in  the  value  of 
the  reactance  according  to  the  value  of 


240 


ALTERNATING  CURRENTS 


The  effect  of  variation  of  the  reactance,  however,  has  already 
been  shown  in  Case  (2),  and  the  deductions  of  Case  (2 ) apply 
to  this  case. 

Case  (4).  Frequency  fixed  and  Voltage  Varied.  — Under  these 
conditions  it  is  obvious  that  the  current  vector  is  fixed  in  phase 
according  to  the  constants  of  the  circuit,  but  its  magnitude 
increases  or  decreases  as  the  voltage  varies. 

Case  (5).  Voltage  and  Frequency  varied  proportionally  to- 
gether. — In  this  case,  if  the  reactance  is  purely  inductive,  the  rise 
of  voltage  matches  the  increase  in  reactance,  and  the  proportion 
of  the  total  volts  absorbed  in  the  reactance  per  ampere  of  cur- 
rent is  unaffected,  hut  the  proportion  of  the  total  volts  absorbed 
in  the  resistance  per  ampere  of  current  by  the  IR  drop  varies 
inversely  with  the  voltage.  The  effect  is  therefore  like  Case 
(1),  in  which  the  reactance  is  constant  and  the  resistance  varies. 
When  voltage  is  zero,  frequency  and  reactance  are  zero,  and 

Z=-^-  = 0,  which  gives  the  same  result  as  I—  — = 0 and  corre- 

sponds  to  the  point  0 of  Fig.  142.  When  voltage  increases  toward 
the  limit  of  infinity,  frequency  and  reactance  increase  toward 
the  same  limit  and  R becomes  of  negligible  influence  on  the 
impedance.  This  condition,  therefore,  gives  the  same  result  as 
E 

I0  = - — — and  corresponds  to  the  point  C in  Fig.  142.  The 

two  circles  of  Fig.  142  correspond  to  F positive  and  2 7t/X 
positive  for  the  upper  half  of  the  right-hand  circle,  F negative 
and  2 irfL  positive  for  the  lower  half  of  the  right-hand  circle, 
E positive  and  2 71 fiL  negative  for  the  upper  half  of  the  left- 
hand  circle,  and  E negative  and  2 7 rfL  negative  for  the  lower 
half  of  the  left-hand  circle.  The  negative  values  for  2 irfL 
may  be  secured  by  substituting  the  effects  of  mutual  induction 
in  the  place  of  self-induction. 

When  capacity  is  associated  with  resistance  either  alone  or 
in  company  with  inductance,  the  reactance  varies  in  a different 
relation  with  the  frequency,  and  the  relationships  of  the  current 
vector  are  less  simple  to  illustrate. 


CHAPTER  V 


THE  USE  OF  COMPLEX  QUANTITIES  EXTENDED 

71.  Vector  Analysis  and  the  Complex  Quantity.  — It  must  be 
continually  borne  in  mind  that  as  far  as  mathematics  are  useful 
to  the  engineer,  or  for  developing  the  applications  and  extend- 
ing our  knowledge  of  discoveries  in  science,  they  must  be  con- 
sidered as  a system  of  logic,  which  may  be  conveniently  applied 
to  the  premises  which  we  possess  for  the  purpose  of  disclosing 
or  predetermining  additional  phenomena.  When  they  are 
used  in  this  manner,  it  is  necessary  for  us  to  obtain  adequate 
physical  conceptions  of  the  results  of  all  mathematical  trans- 
formations. Our  ordinary  systems  of  mathematics  are  based 
upon  certain  assumed  premises  called  axioms,  and  it  is  clear 
that  if,  by  abandoning  these  axioms  and  accepting  other  and 
different  premises  which  we  may  or  may  not  call  axioms,  we 
can  thereby  obtain  a more  convenient  and  satisfactory  system 
of  logic  to  apply  to  our  ends,  then  we  are  justified  in  abandon- 
ing the  old  and  adopting  the  new  or  of  adding  the  new  to  the 
old.  This  is  what  we  do  to  a certain  extent  when  we  add  the 
processes  of  vector  analysis  to  the  processes  of  Euclidian  and 
Cartesian  geometry  for  the  solution  of  alternating-current 
problems.  We  do  an  analogous  thing  when  we  abandon  the 
ordinary  geometry  and  adopt  Hamilton’s  quaternion  methods 
or  Grasmann's  powerful  space  analysis  for  treating  physical 
problems  involving  vectors  in  space.  It  is  just  as  irrational 
for  us  to  confine  our  attention  to  the  mathematics  of  a line,  as 
is  done  in  ordinary  algebra,  when  we  can  usefully  extend  our 
consideration  to  an  entire  plane  or  to  space,  as  it  was  absurd 
for  the  early  algebraists  to  neglect  all  but  positive  roots  in 
their  solutions  of  equations.  The  old-time  axioms  might  still 
be  conclusive  if  viewed  within  the  narrow  horizon  of  Euclid’s 
time,  but  in  our  wider  horizon  they  extend  to  but  a small  por- 
tion of  the  range  of  our  observations  and  we  can  properly 
extend  our  methods  of  reasoning  to  correspond  with  the 
borders  of  our  knowledge. 

o 

k 241 


212 


ALTERNATING  CURRENTS 


Until  the  engineer  has  learned  that  the  place  of  mathematics 
in  his  professional  training  is  no  other  than  that  of  a system 
of  logic  and  therefore  a tool,  it  is  unsafe  for  him  to  use  anv 
form  of  mathematics  higher  than  arithmetic  and  the  elements 
of  ordinary  algebra ; but  when  it  is  recognized  that  mathe- 
matics compose  a system  of  logic  and,  indeed,  a system  of  the 
most  powerful  kind  which  may  be  used  in  improving  engineer- 
ing processes,  the  propriety  of  the  use  of  various  branches  of 
higher  mathematics  in  engineering  studies  is  manifest. 

In  Art.  6 the  following  forms  and  conditions  have  been  given 
for  the  vector  or  complex  quantity  : 

A = a + jb, 

A — ra( cos  0 + j sin  0)  = me30,* 

where  m is  the  Tensor  and  0 the  Argument  of  the  vector 
quantity  and  a and  b are  the  lengths  of  rectangular  com- 
ponents of  the  vector.  Also, 

m2  = a2  + b2 

, , b sin  0 

a cos  0 

In  these  expressions  the  scalar  values  of  A and  m are 
equal,  hut  the  former  is  a vector  having  direction,  while  m 
may  be  considered  as  a simple  number.  The  expression 
(cos  8 4 - j sin  6)  = ej0  is  sometimes  called  the  Direction  coeffi- 
cient or  Versor  of  the  expression  of  which  it  is  a part.  It  may 
be  abbreviated  into  (cjs)  0.  The  tensor  (also  sometimes  called 
the  modulus)  of  the  vector  represents  its  actual  numerical 
value  and  it  is  therefore  a “ scalar  ” quantity  ; that  is,  it  is  a 
quantity  which  has  numerical  or  scale  value  only,  without  fixed 
direction.  It  is  by  reason  of  the  latter  fact  that  it  is  some- 
times called  the  “ absolute  value  ” of  the  vector.  The  versor 
of  a vector  has  direction  fixed  by  the  argument  8 , but  is  of 
unit  length  only.  The  product  of  versor  by  tensor  therefore 
fixes  the  vector  in  terms  of  both  length  and  direction. 

The  operator  j and  the  coplaner-vector  quantities  treated  in 
this  hook  are  assumed  to  be  subject  to  the  usual  algebraic  laws, 

* See  Williamson’s  Differential  Calculus,  pp.  60-69. 


THE  USE  OF  COMPLEX  QUANTITIES  EXTENDED  243 


like  the  law  of  signs,  the  law  of  indices,  and,  in  general,  the 
commutative,  associative,  and  distributive  laws.* 

Such  characteristics  have  been  given  the  operator  / that  the 
imaginary  quantity  V — 1 may  be  substituted  for  it,  and  the 
equation 

a -\-jb  = a + V — 1 b 

holds  true;  for  /2  = — 1 by  definition.  Hence  / = V—  1. 1 
The  symbol  V—  1 may  therefore  be  considered  to  have  a physi- 
cal meaning  like  that  of  the  operator/,  and  if  it  is  met  with  in 
the  roots  of  an  equation  or  elsewhere,  it  may  be  replaced  by  /. 
When  interpreted  in  this  way,  it  is  to  be  observed  that  an 
“ imaginary  ” expression  assumes  its  true  significance  of  a vec- 
tor quantity  having  a numerical  magnitude  and  definite  direc- 
tion. The  real  roots  of  an  equation  or  any  other  “real” 
expression  may  similarly  be  considered  as  vector  quantities,  all 
of  which  lie  on  the  same  line  which  is  at  right  angles  to  the 
axis  of  “imaginaries.” 

Since  /2=— 1,  — /2=+l,  from  which  follows  the  relation 
^ = — / ; or,  in  words,  the  effect  of  applying  the  reciprocal  of 

the  operator  to  a vector  quantity  is  the  same  as  the  effect  of 
applying  the  reversed  operator. 

In  general  it  must  be  remembered  that  any  direction  is  rep- 
resented equally  by  d,  0 + 2 7r,  6 + 4 tt,  etc.,  to  an  infinite 
number  of  angles,  so  that  the  argument  of  the  vector  has  many 
values,  except  where  the  conditions  specially  fix  it.  We  shall, 
as  a rule,  consider  it  for  convenience  as  equal  to  0 even  where 
it  is  not  fixed  by  the  conditions;  but  where  distinct  generaliza- 
tion of  the  argument  is  desirable,  we  shall  write  it  0 + 2 7 rn. 
The  complex  quantity  which  has  been  developed  lies  in  one 
plane  and  its  particularly  marked  simplicity  is  a result  of  its 
uniplaner  condition. 

Coplaner  (uniplaner)  vectors,  with  their  corresponding  com- 
plex representations,  are  quite  useful  in  representing  the  rela- 
tions arising  in  problems  which  are  caused  by  current  flow  in 
alternating-current  circuits.  These  problems  deal  with  periodic 
quantities  which  may  be  treated  as  though  they  were  the  result 

* For  a discussion  of  these  laws  in  relation  to  vector  algebra,  see  Hayward’s 
Vector  Algebra  and  Trigonometry , pp.  2,  14,  and  39. 

t Art.  6. 


244 


ALTERNATING  CURRENTS 


of  uniform  rotating  vectors,  or  stationary  vectors  in  a plane 
which  itself  rotates  about  a perpendicular  axis.  The  methods 
used  in  vector  algebra  are  analogous  to  those  which  are  so 
largely  used  in  the  graphical  solutions  relating  to  the  composi- 
tion and  resolution  of  forces  in  graphical  statics  and  to  the 
composition  and  resolution  of  velocities  and  moving  forces  in 
graphical  dynamics.  The  alternating-current  vectors  are  mov- 
ing vectors,  and,  consequently,  the  methods  must  be  similar  to 
those  used  in  the  vector  processes  applied  to  dynamics. 

72.  Addition  and  Subtraction  of  Complex  Expressions.  — Ad- 
dition and  subtraction  of  these  expressions,  as  has  already  been 
indicated,*  is  accomplished  as  in  ordinary  algebraic  operations; 
as,  for  instance, 

O +jb)  + 0 —jd)  = a + c +j  (b  - d), 

(a  +jb)  - (c  —jd)  = a — c +j(b  + d). 

The  rules  for  addition  are,  — reals  must  be  added  numerically 
to  reals,  and  imaginaries  must  be  added  numerically  to  imagi- 
naries.  Whence,  when  addition  is  performed,  (1)  the  horizontal 
component  of  the  resulting  vector  is  equal  to  the  sum  of  the 
horizontal  components  of  the  vectors  added;  (2)  the  vertical 
component  of  the  resulting  vector  is  equal  to  the  sum  of  the 
vertical  components  of  the  vectors  added. 

Subtraction  being  the  reverse  process  gives:  (1)  the  hori- 
zontal component  of  the  resulting  vector  is  equal  to  the  differ- 
ence between  the  horizontal  components  of  the  minuend  and 
subtrahend ; (2)  the  vertical  component  of  the  resulting  vector 
is  equal  to  the  difference  between  the  vertical  components  of 
the  minuend  and  subtrahend. 

In  case  an  expression  has  only  a horizontal  or  only  a vertical 
component,  then,  of  course,  the  addition  or  subtraction  is  carried 
on  as  though  the  lacking  component  were  equal  to  zero. 

73.  Multiplication  and  Division  of  Complex  Expressions.  — 
The  process  of  multiplication  or  division  of  a complex  by  a 
scalar  quantity  (that  is,  a quantity  which  has  only  a numerical 
value)  is  similar  to  the  corresponding  ordinary  arithmetical 
process  of  multiplication  or  division.  For  instance, 

(a  +jb)  x 2 is  2 (a  +jb)  = 2 a + j 25 
and  (a  +jb)  -r2is|(a  +jb)  = | a +j  b. 


* Art.  8. 


THE  USE  OF  COMPLEX  QUANTITIES  EXTENDED  245 


A.' 

k/ 

i 

kV  i 

<— 1_ _J 

This  is  illustrated  in  Fig.  144  where  OA  is  a vector.  If  this 
vector  is  multiplied  by  2,  its  direction  is  not  changed,  but  its 
length  is  doubled  and  the  vector  becomes 
OA'.  The  proportions  of  the  vector  show 
plainly  that  the  horizontal  component  of 
OA'  is  equal  to  2 a and  the  vertical  com- 
ponent is  equal  to/2  6.  If  OA  is  divided 
by  2,  it  becomes  a vector  of  | the  length 
and  the  same  direction  as  before,  and  is 

represented  by  the  vector  0A1.  The  pro-  q a X 

portions  of  the  vector  again  show  clearly  Fig.  144.— Graphical  Rep- 
that  the  horizontal  component  now  is  Quantity  multiplied  or 
equal  to  \a  and  the  vertical  component  divided  by  a Scalar  Quan- 
is  equal  to  j\  b.  tlty- 

Algebraical  processes  also  immediately  apply  in  the  multipli- 
cation or  division  of  complex  quantities  by  complex  quantities, 
as  will  be  more  fully  illustrated.  W e will  suppose  that  the  two 
complex  quantities  a + jb  and  c + jd  are  to  be  multiplied 
together  and  that  these  quantities  are  equal,  respectively,  to 
mx  (cos  6X  A / sin  dx)  = and  m2  (cos  d2  + j sin  d2)  = m2ejB\ 

Multiplying  the  exponential  forms  of  the  two  expressions 
together  gives  as  a result 

m1m2ej{ei  + e*)  = mxm2  [cos  (6X  + d2)  +/  sin  (6X  + d2)]. 

If  we  directly  multiply  together  the  expressions 

m^cos  6X  +j  sin  0X)  and  w2(cos  d2  +/  sin  d2), 


we  immediately  come  to  the  expression 

m1m2[cos  (dj  + d2)  +j  sin  (d1  + d2)]. 

We  therefore  have  a satisfactory  demonstration  that  the  ordi- 
nary process  of  multiplication  in  algebra  applies  to  the  multi- 
plication of  these  coplaner-vector  expressions.  Consequently, 

(a  +/6)(c+/d)  must  be  equal  to  ac  A j ad  A job  Aj^bd, 

which,  again,  is  equal  to 

ac  — bdA  j(bc  + ad'). 

This  shows  that 

ac  — bd  is  equal  to  mxm2  cos  (dx  -f  d2) 
be  + ad  is  equal  to  mxm2  sin  (0X  -}-  d2). 


and 


246 


ALTERNATING  CURRENTS 


These  equalities  may  also  be  proved  by  expanding  the  expres- 
sions mpn2  cos(d1  -f  $2)  and  m1w2sin(d1  + d2),  and  assigning  to 
mx  sin  0V  m1  cos  6V  m2  sin  d2,  m2  cos  d2  their  respective  values  in 
terms  of  a , b , c,  and  d,  indicated  by  Fig.  145. 

By  an  inspection  of  the  trigonometrical  form  of  the  foregoing 
product,  it  is  made  evident  that  the  product  of  two  complex 
quantities  is  another  complex  quantity  in  which  the  tensor  or 

modulus  is  equal  to  the  product  of  the 
tensors  of  the  multiplier  and  multipli- 
cand,, and  in  which  the  argument  is 
equal  to  the  su?n  of  the  arguments  of 
multiplier  and  multiplicand  (see  Fig. 
145)  ; also  that  the  product  vector 
has  a horizontal  component  which 
is  equal  to  the  difference  between 
the  products  of  the  horizontal  com- 
ponents and  the  vertical  compo- 
nents respectively  of  multiplier  and 
multiplicand,  and  a vertical  compo- 
nent which  is  equal  to  the  sum  of 
the  product  of  the  horizontal  com- 
ponent of  the  multiplier  with  the  vertical  component  of  the 
multiplicand  and  the  product  of  the  vertical  component  of  the 
multiplier  with  the  horizontal  component  of  the  multipli- 
cand. 

By  Complementary  vectors  are  meant  vectors  with  equal  hori- 
zontal components,  equal  vertical  components,  and  equal  but 
reversed  arguments.  Thus,  if  we  desire  to  multiply  a 4 -jb  btT 
a—jb  (i.e.  a vector  quantity  by  its  Complement),  we  get  the 
following  result, 

(a  -f/4)(a  — jb')  = a2  — jab  +jab  — j2b2  = a2  + b2  = m2\ 


0 a 0 

Fig.  145.  — The  Product  of  Two 
Vectors  OA1  and  OA2  is  OR,  in 
which  OR  has  a length  equal  to 
the  length  of  OAi  multplied  by 
the  length  of  OA.2. 


and  again 

(a  +i^)(dr  — jb)  = m 2 [cos  {6  — @)  + j sin  (d  — d)]  = m2. 

The  product  of  a vector  and  its  complement  therefore  is  a vector 
with  zero  argument  and  may  be  considered  as  an  algebraic  value 
equal  to  the  sum  of  the  squares  of  the  two  components  of  either  of 
the  complementary  vectors. 

In  division  a similar  process  may  be  followed.  We  have 


THE  USE  OF  COMPLEX  QUANTITIES  EXTENDED  247 


(a  +jh)  -h  (<?  - \-jd ) = mff01  -j-  mff02  = -1  [V(0l_9a)] 

m2 

= ^ [cos  (0j  — 02)  4-  j sin  (^01  — Of] . 
w2 

The  demonstration  of  this  follows  in  a manner  identical  with 
the  demonstration  for  multiplication  given  above.  The  ratio 
of  two  vectors  is  therefore  another  vector  in  which  the  tensor 
is  equal  to  the  ratio  of  the  tensors  of  dividend  and  divisor  and  the 
argument  is  equal  to  the  difference  of  the  arguments  of  dividend 
and  divisor. 

Also  multiplying  both  numerator  and  denominator  by  (<?  — jd~) 
and  remembering  that  — j2  — + 1,  there  results 

a +jb  _ (a  + jh)(c  —jd) 
c +jd  c2  + d2 

The  denominator  in  this  case  is  a real  or  arithmetical  quantity 
and  the  numerator  is  equal  to  the  product  of  two  vectors.  Ex- 
panding shows  that  the  above  ratio  is  equal  to 


whence, 

and 


ac  + hd  +j(hc  — ad ) _ac  + hd  , .he  — ad  m 
c2  + (p  “ c2  + d2  +J  c2  + d2  ’ 

m,  .a  a N ac  + hd 


— 1 sin  (6l  — Of  — 


he  — ad 
c2  + d2 


Since  m2  is  numerically  equal  to  the  square  root  of  e2  + d?,  it  is 
clear  that 

ac  -f  hd 


and 


mx  cos(dj  — Off  - 
m1sin(d1  — 0f  = 


(c2  -f  d2)  - 
he  — ad 
O2  + d2ff 


The  foregoing  theorems  of  multiplication  and  division  are 
equally  applicable  to  stationary  vectors  and  to  rotating  vectors 
of  equal  frequency.  The  multiplication  together  of  rotating 
vectors  of  unequal  frequencies  gives  a different  result  for  the 
reason  that  Qx  and  02  do  not  increase  at  the  same  rate.  Since 


sin  ma  sin  nada  = 0 when  m and  n are  unequal  integers,  it  is 


obvious  that  the  product  of  a rotating  vector  by  a rotating 


248 


ALTERNATING  CURRENTS 


vector  having  a different  frequency  which  is  any  integral  num- 
ber of  times  the  frequency  of  the  first  vector,  is  equal  to  zero. 
This  also  follows  from  the  usual  theorems  of  the  products  of 
simple  harmonic  motions.  Also,  since 


f 


sin  ma  sin  nada  = - 

9 


sin  (m  — n)  a sin  (m  + ri)  a 
m — n m -\-n 


it  is  obvious  that  the  product  of  the  two  rotating  vectors  does 
not  reduce  to  zero  in  one-half  period  in  case  n is  a mixed 
number  instead  of  an  integer,  hut  the  product  does  reduce  to 
zero  in  the  number  of  half  periods  equal  to  the  reciprocal  of  the 
fractional  part  of  n.  The  product  of  two  rotating  vectors  of 
different  frequencies,  therefore,  always  reduces  to  zero  in  some 
definite  number  of  half  periods,  although  it  may  have  a large 
finite  value  for  some  particular  half  period. 

74.  Reciprocal  of  a Vector  Expression.  — The  reciprocal  of  a 
vector  may  be  written  this  way  : 


1=  1 ^ 1 _ 1 _lc-rt 

/[  a+jb  ra(cos  6 +j  sin  6)  meje  m 

= — [cos  ( — 0)  +/  sin  ( — 0)] . 
m 

Also  since 

a — jb  = m (cos  6 —/sin  6)  = m [cos(—  0)+j  sin(—  #)], 

it  is  clear  that  -. 

1 _a  — jb 

a 4 -jb  m2 


This  may  also  be  proved  by  rationalization.  Thus,  multiplying 
the  numerator  and  denominator  by  a — jb  gives 


a — jb  _ a — jb 
(■ a 4 -jbj(a  —jb)  a 2 4-  b 2 


a — jb  _ a 

9 r 9 

m*  nv* 


Hence,  the  reciprocal  of  a vector  is  a vector  whose  tensor  is  equal  to 
the  reciprocal  of  the  original  tensor , and  whose  argument  is  equal  to 
the  argument  of  the  original  vector  with  reversed  sign. 

The  horizontal  component  of  the  reciprocal  is  equal  to  the  hori- 
zontal component  of  the  original  vector  divided  by  the  square  of 
the  tensor , and  the  vertical  component  of  the  reciprocal  is  equal  to 
the  vertical  component  of  the  original  vector  with  reversed  sign  and 
divided  by  the  square  of  the  tensor  of  the  original  vector. 


THE  USE  OF  COMPLEX  QUANTITIES  EXTENDED  249 


75.  Involution  and  Evolution  of  Complex  Quantities. — The 

square  of  the  vector  A is 

A 2 = (me?0)2  = m2e,2e  = m2( cos  2 0 +j  sin  2 0). 

This  comes  from  the  rules  of  multiplication  given  above ; or, 
the  square  of  a vector  has  a tensor  equal  to  the  square  of  the  origi- 
nal tensor  and  an  argument  equal  to  twice  the  original  argument. 

Other  powers  of  vectors  follow  the  same  laws,  which  come 
directly  from  the  rules  of  multiplication.  For  instance, 

(a  + jby=  [m(cos  6 + j sin  0)]”=  m”eine  — m"( cos  n0  + j sin  n0f. 

If  the  exponent  n is  negative,  its  sign  is  reversed  in  the 
above  expression  as  compared  with  the  expression  when  n is 
positive.  We  then  have  the  reciprocal  of  the  tensor  and  the 
reversed  angle. 

Also 

1 i i .0.  if  0 , . Q\ 

( a + jb )"  = [m(cosd+y  sin  0)]  ”=  m n ” =mn(  cos  -4 -j  sm  - j- 

From  this  it  can  be  seen  that  evolution  is  clearly  a process 
of  multiplication  and  that  involution  may  be  looked  upon  either 
as  a process  of  multiplication  or  a process  of  division. 

It  has  been  previously  shown  that/-1  =-  — — / ; that  is,  /-1 

3 

is  an  indicator  of  rotation  backwards  by  the  amount  of  one 
quadrant.  In  the  same  way  jn  indicates  a rotation  forward  by 

n quadrants  and /“”  = •“=— /"  indicates  rotation  backwards 

by  n quadrants. 

In  case  the  vector  a +jb  is  squared,  we  have 
(a  +jbf  = m2( cos  2 0 + j sin  2 0), 
and  this  is  also  equal  to 

a2  + / 2 ab  +j2b2  = a2  — b2  +j  2 ah , 
and  therefore 

to2  cos  2 0 — a2  — b2 
and  m2  sin  2 0 — 2 ab. 

The  square  of  a vector  therefore  has  a horizontal  component 
equal  to  the  difference  of  the  squares  of  the  horizontal  and,  vertical 
components  of  the  vector , and  a vertical  component  which  is  equal 
to  twice  the  product  of  the  horizontal  and  vertical  components  of 
the  vector. 


250 


ALTERNATING  CURRENTS 


In  taking  the  root  (such  as  the  nth  root)  of  a vector,  we  get 

the  product  of  the  given  root  of  the  tensor  and  cos  -+;  sin  ", 

n n 

That  is,  the  tensor  of  the  root  of  a vector  is  equal  to  the  real 
algebraic  root  of  the  tensor  of  the  vector.  But  the  root  of  a 
vector  has  direction  of  its  own  which  depends  on  the  argument 
of  the  original  vector.  The  directions  of  the  root  vectors  are 


6 

given  by  -,  which  affords  n different  directions.  For  instance, 
let  0 = 30°  and  n = 2 ; then  | = 15°,  ^+-360°  = 195°  6 + 720 ° 


2 


2 


9 


= 375°  = 15°  etc.,  giving  just  two  roots  which  have  opposite 
directions.  Again,  let  0 = 30°  and  n = 5 ; then, 


0 _ ao  o + 360° 
5 ’ 5 

0 4- 1440° 
5 


= 78o  0 + 720°  _ 15QO  0 + 1080° 

' 5 ’ 5 


_ 294°,  fl+  1800°: 


6°,  etc.. 


999° 


giving  five  different  directions  for  the  five  roots. 

Again,  let  0=0,  and  n — 2,  then  ^=0°,  = 180°, 

0 4-  790°  ~ 2 

— — = 360°  = 0°,  etc.,  which  shows  that  the  square  roots  of 

Z 

a real  quantity,  that  is,  a quantity  with  a zero  argument,  are 
both  real  quantities  (that  is,  they  lie  in  the  horizontal  axis  of 
reals)  and  are  of  opposite  signs.  Again,  letting  0 = 0,but  taking 


n = 4,  gives 


0 = ^ 0 + 360°  = 9QO  0 + 720°  = 18()0  0 + 1080° 
4 4 4 4 

= 270°,  ? + 144Q°  =360°  = 0°,  etc., 

4 


showing  that  the  fourth  roots  of  a real  quantity  are  in  opposite 
pairs,  of  which  one  pair  are  reals  and  the  other  pair  pure 
imaginaries. 

Following  up  in  this  manner  our  rule  for  evolution,  it  will  be 
observed  that  the  »th  root  of  any  vector  quantity  has  a tensor  of 
fixed  value  equal  to  the  nth  root  of  the  original  tensor,  but  that 
there  are  in  fact  n root  vectors,  each  with  a direction  of  its  own. 

l 

The  n roots  of  (1)"  are 


vt(> 


2 t rA  . . ■ 2 irk 

cos V j sm  — 

V n n 


_ . ink 


THE  USE  OF  COMPLEX  QUANTITIES  EXTENDED 


251 


d (OA) 


where  k is  to  be  given  the  values  0,  1,  2,  3 •••  n — 1,  taken  in 
order.  In  this  case  (1)  is  considered  to  be  a vector  lying  along 
the  horizontal  axis  of  reals,  and  therefore  6 = 0.  Since  the 

nth  root  of  (1)  is  a vector  with  argument  it  is  clear  that 

1 . n 27T 

multiplying  a vector  by  (l)re  causes  rotation  by  an  angle  - — • 

n 

We  have  here  a justification  of  our  use  of  the  imaginary  which 
enables  us  to  determine  all  the  roots  of  an  equation.  It  would 
be  just  as  irrational  to  neglect  the  imaginary  roots  as  it  was  in 
the  early  algebraists  to  also  neglect  the  negative  roots  and  con- 
sider only  the  positive  roots  as  of  any  significance. 

76.  Differentiation  and  Integration  of  Complex  Quantities.  — 
Differentiation  of  a vector  quantity  d(A ) ~ d(a  + jb)  becomes, 
in  conformity  with  the  distributive 
law  of  algebra,  the  same  as  da  -+- 
d(jb),  and  this  becomes  in  conformity 
with  the  commutative  law  the  same 
as  da  + jdb.  The  j here  acts  like 
any  constant  quantity  in  an  equation 
which  is  differentiated  and  serves  as 
an  operator  or  indicator  to  show 
that  db  is  measured  vertically.  We 
can  illustrate  the  condition  by  Fig. 

146  in  which  OA  represents  the  vec- 
tor and  AA'  may  be  considered  to  be  its  increment.  AA' 
therefore  is  equal  to  d(  OA'),  and  da  and  jdb  are  respectively 
the  horizontal  and  vertical  components  of  the  increment. 

It  is  evident  that  an  integration  performed  on  the  expression 
da  + jdb  must  result  in  a +jb , provided  the  above  reasoning  is 
correct. 

Suppose  the  process  of  differentiation  is  applied  to  the  ex- 
pression m( cos  6 +j  sin  6).  We  then  have 

d[m(cos  6 +j  sin  #)]  = dm( cos  6 +j  sin  6)  + md( cos  6 +j  sin  6) 
= dm  (cos  6 +j  sin  6)  + m(j  cos  6 — sin  6)dd. 

Now  writing  (cjs)d  for  the  direction  coefficient  cos  6 +j  sin  6, 
the  foregoing  becomes 

(cjs  )6dm  +/m(cjs)ddd  = (dm  -\-jmdd)  (cjs)0. 

This,  then,  is  the  differential  of  the  expression  a +jb.  If  the 


0 « 

Fig.  116.  — Illustration  of  a Vector 
increased  by  an  Increment. 


252 


ALTERNATING  CURRENTS 


expression  is  in  the  form  meje  and  we  differentiate  it,  we  have 
this  result, 

(e'j6')dm  -f  jmeJ0dO  = (dm  +jmdd)e36. 

It  will  also  be  observed  from  this  that  the  real  part  of  the 
differential  is 

da  = cos  Odm  — m sin  QdO  ; 
and  the  imaginary  part  is 

jdb  = j (sin  Odm  + m cos  0O0). 

We  have  written  (cjs)d  for  the  expression  cos  0 +j  sin  0 in 
the  preceding  paragraphs.  The  expression  (cjs)  may  be  pro- 
nounced as  though  it  were  spelled  “ siss,”  and  it  can  be  readily 
seen  that  it  is  derived  by  combining  the  initial  of  cos  with  the 
operator  j and  the  initial  of  sin  in  the  expression  cos  0+j  sin  0. 
This  abbreviation  is  a convenient  one  and  will  be  used  here- 
after. It  should  be  remembered  that  (cjs)#  = ej0. 

77.  Logarithms  of  Complex  Quantities.  — In  the  preceding 
portion  relating  to  the  differentiation  of  complex  quantities, 
we  have  shown  that  where  A is  a complex  quantity  equal  to 
?rc(cjs)#,  then  dA  = (dm  +jmdO) (cjs) 0 ; and  from  these  we 
cl  cl/TCL  * 

get  = \-jdO,  the  process  performed  having  been  to 

A m 

divide  the  vector  into  its  differential.  Since  the  natural  loga- 
rithm of  any  quantity  is  by  definition  equal  to  the  integral  of 
the  ratio  of  the  differential  of  a quantity  to  that  quantity,  that 

is,  log  , it,  follows  that  log  A = j"—  + J'jdO.  In- 

tegrating these  gives,  log  A = log  m +j(0  + 2 7 to),  where  n is 
0 or  an  integer.  The  term  2 7 to  in  the  final  expression  may  be 
looked  upon  as  a constant  of  integration,  and  it  will  be  noticed 
that  its  addition  to  0 does  not  change  the  real  direction  of  the 
vector,  but  shows  that  the  angle  0 is  not  iixed  except  in 
special  cases  and  that  the  angle  0 may  have  an  infinite  num- 
ber of  values  differing  from  each  other  by  2 7 -j  or  360/°. 
These  expressions  show  that  the  logarithm  of  a complex  num- 
ber depends  upon  the  logarithm  of  the  tensor  of  the  number 
and  upon  its  argument,  and  that  any  function  may  have  an 
infinite  number  of  logarithms  which  arise  in  an  arithmetical 
series  with  the  difference  equal  to  2 7r/. 


THE  USE  OF  COMPLEX  QUANTITIES  EXTENDED 


253 


It  is  easy  to  get  from  this  the  logarithm  of  the  ratio  Ax  to 
A9  ; thus, 


log  = log  Ax  - log  A2  = log  mx  + jd i - log  m2  - j02 

A0 


= log^l  + jQ01  - 02). 


m„ 


In  the  same  manner  log  (A.J  A2)  may  be  shown  to  be  equal  to 
log  mxm2  +j(0x  + 02). 

78.  Combination  of  Admittances.  — In  the  previous  chapters, 
series  circuits  have  been  dealt  with  in  most  instances.  The 
following  pages  of  this  chapter  will  deal  more  particularly  with 
circuits  in  parallel  and  series-parallel. 

If  impedances  are  in  parallel,  their  reciprocals  must  be  com- 
bined, in  which  case  the  resultant  is  the  reciprocal  of  the 
impedance  of  the  divided  circuit.  It  is  evident  that  the  com- 
ponents of  the  admittance  (reciprocal  of  impedance)  will  not  be 
equal  to  the  reciprocals  of  the  components  of  the  impedance, 
but  they  may  be  found  in  terms  of  the  impedance  components, 
as  pointed  out  in  Arts.  63  and  74. 

The  component,  g,  of  the  admittance  that  lies  along  the  hori- 
zontal axis  is  the  component  termed  the  Conductance  and  the 
vertical  component,  5,  the  Susceptance.  The  latter  is  positive 
in  direction  if  caused  by  capacity,  and  negative  if  caused  by 
self-inductance.  If,  then,  g , 5,  are  the  admittance  components 
(conductance  and  susceptance)  and  R,  X , the  impedance  com- 
ponents (resistance  and  reactance)  of  a single  circuit,  we  may 
write  by  the  principles  of  geometric  multiplication  above 
stated,  1 


Y=  — = g Tjb  = 


R±jX 


O) 


The  numerical  values  of  the  first  and  second  terms  of  the  right- 
hand  member  of  the  expression  Y = g T jb  are  respectively 
proportional  to  the  active  current  and  quadrature  current  in  a 
circuit.  When  a circuit  contains  inductive  reactance  only,  X 
is  essentially  positive,  but  the  quadrature  current  lags  90°  behind 
the  active  current,  so  that  b is  essentially  negative.  When  a 
circuit  contains  capacity  reactance  only,  X is  essentially  nega- 
tive, but  the  quadrature  current  leads  the  active  current  by  90°, 
so  that  b is  essentially  positive.  When  a circuit  contains  both 


254 


ALTERNATING  CURRENTS 


inductive  and  capacity  reactance,  the  signs  of  X and  b are  depend- 
ent upon  the  relative  magnitudes  of  the  inductance  and  capacity. 
To  reduce  the  equation 


9 Tjb  = 


1 

R±jX 


(&) 


to  a more  convenient  form,  the  numerator  and  denominator  of 
the  right-hand  member  may  be  multiplied  by  R T jX ; whence 


R t jX  RTjX 

(. R T jX)  ( R ± jX)  R*  + X2‘ 


since  y2  indicates  the  operation  which  is  equivalent  to  multiply- 
ing by  — 1.* 


Hence, 


9^  jf>  = 


R 


R 2 + X2  T J R2  + X2  ’ 


X 


but  the  numerical  value  of  the  impedance  is 


VR Tx2. 

Therefore  g T jb  = T 3 

R ii , ^ 

or  9 = y*  and  b = 


If  R , X,  and  Z are  known  or  can  be  determined  from  the  con- 
ditions presented,  problems  relating  to  parallel  circuits  can  now 
be  solved. 


Fig.  147.  — Admittances  in  Parallel. 


If  in  Fig.  147  g\  b'  and  g",  b"  are  the  components  of  the 
admittances  of  two  parallel  circuits  having  impedances  Z' 
and  Z" , 


*Art.  8. 


THE  USE  OF  COMPLEX  QUANTITIES  EXTENDED  255 


g'-jv 


-EL 

z ,2  3 Z'2 


and 


„ , R"  , -X" , 

^ ^ ^//2  ^7/2  1 


and  if  g and  £ are  the  components  of  the  total  admittance, 
Y = g + jb  = g'  + g"  + j (b" 


R'  , R"  , .X"  .X' 

7 Z'2  Z"2  ^ Z"2  ' Z'2 


The  intrinsic  sign  of  jb  depends  upon  the  relative  signs  and 
X'  X" 

magnitudes  of  and  The  impedance  of  the  circuit  is  the 


reciprocal  of  the  admittance  thus  found. 

These  processes  which  enable  us  to  find  the  joint  impedance 
and  admittance  of  parallel  or  series  circuits  when  the  elements 
of  the  individual  parts  of  the  circuits  are  known,  equally  enable 
us  to  find  the  impedance  and  admittance  of  any  combination  of 
such  circuits  by  computations  which  are  almost  as  simple  and 
rapid  as  those  which  would  be  used  in  dealing  with  a direct- 
current  system.  Also,  when  the  impedances  of  any  combina- 
tion of  circuits  have  been  obtained,  it  is  possible  to  find  the 
voltages  in  any  portion  when  a sinusoidal  current  is  flowing 
and  to  find  the  current  when  a sinusoidal  voltage  is  applied. 

The  meaning  of  the  terms  in  the  expression  for  impedance 
and  admittance  may  be  explained  by  multiplying  (i2  ± jXj  by  1 
(current  in  the  circuit),  when  it  is  evident  that  rl  is  the  active 
voltage  and  XI  the  component  of  voltage  acting  against  the 
reactance;  and  by  multiplying  (g  T jbj  by  E (voltage  im- 
pressed on  the  circuit),  when  gE  is  the  active  component  of 
the  current  and  bE  the  quadrature  current. 

The  following  is  a recapitulation  of  the  formulas  for  the 
analytical  solution  by  geometric  processes  of  many  problems 
relating  to  alternating-current  circuits.  The  use  of  a vinculum 
over  a letter  (as  Z ) indicates  its  value  as  a complex  quantity 
and  the  letter  without  the  vinculum  indicates  the  tensor  or 
numerical  value.  In  this  recapitulation  the  small  letters  r,  x,  y, 
and  z represent  the  resistance,  reactance,  admittance,  and  im- 
pedance of  parts  of  the  circuit  while  the  capitals  i?,  X , IT,  and 
Z represent  the  same  quantities  for  the  whole  circuit.  In  the 
same  manner  the  capitals  Cr  and  B stand  for  circuit  parts  and 
the  small  letters  g and  b for  the  whole  : 


256 


ALTERNATING  CURRENTS 


Geometric  Equations 

z = r ±jx. 

Z — IjZ  — ±j"Lx 

= ^ ±y* 

= Z (cos  0 +y  sin  6). 

y = l = aTjB 


r .x 
— ~2  “2‘ 


Y=^y  = Hr-Tj^\ 
z2  z2 

_rtx 

Z2  Z2 

y E - 
I=2  = *r- 

e = iz  = £- 

Y 


Algebraic  Equations 

g = Si=RYM  = ^i=XYK 


z=  VW+X2  = 


Y = 


V^2  + b2  — 


n 

cos  0 
9 _ 


X 

sin  6 
b 


cos  6 sin  6 


tan  0 = — = -■ 
R g 

T=i=£Y- 


The  currents  and  voltages  in  the  geometric  equations  are 
true  vector  quantities  with  magnitudes  and  relative  directions. 
The  impedances  and  admittances  are  of  the  character  of  vector 
operators. 

It  will  be  observed  that  when  E is  computed  by  means  of 
the  last  equation  in  the  column  of  geometric  equations,  its 
phase  is  measured  with  respect  to  the  current ; and  when  I is 
computed  by  means  of  the  next  to  the  last  equation  in  the 
column  of  geometric  equations,  its  phase  is  measured  with 
respect  to  the  phase  of  the  voltage.  The  two  measurements 
give  the  same  angle  but  it  is  taken  in  opposite  directions. 


CHAPTER  VI 


SOLUTION  OF  CIRCUITS — APPLICATION  OF  GRAPHICAL 
AND  ANALYTICAL  METHODS 


79.  Graphical  Methods.  — As  previously  pointed  out  graph- 
ical methods  often  lend  themselves  satisfactorily  to  the  solution 
of  problems  relating  to  circuits  upon  which  sinusoidal  voltages 
are  impressed.  This  application  of  graphical  processes  was  first 
brought  to  general  attention  by  T.  H.  Blakesley.*  This 
chapter  is  a general  review  of  portions  of  chapters  I and  V, 
but  it  goes  further,  dealing  extensively  with  circuits  in  parallel 
and  with  complex  combinations  of  series  and  parallel  circuits. 
It  also  develops  a number  of  important  general  relations  which 
have  heretofore  not  been  dealt  with. 

To  go  more  into  detail  concerning  the  graphical  combination 
of  vectors  than  lias  been  done  previously,  suppose  the  line  OA 
in  Fig.  148  is  conceived  as  swinging  at  a uniform  angular 
velocity  oo  around  the  point  0,  the  angle  « which  it  makes 
with  the  horizontal  axis  OX  at 
any  instant  is  a = cot,  where  t 
is  the  interval  of  time  during 
which  the  line  describes  the 
angle  a.  The  instantaneous  pro- 
jection Oa  upon  the  vertical  axis 
OY,  of  the  line  OA,  has  a value 
Oa  — OA  sin  a.  If  OA  is  propor- 
tional to  the  maximum  value  of  a 
sinusoidal  function,  its  instantane- 
ous values  are  proportionally  represented  by  the  instantaneous 
projections  of  OA ; and  if  OA  is  proportional  to  the  effective 
value  of  a sinusoidal  function,  the  instantaneous  values  of  the 
function  are  proportionally  represented  by  the  product  of  V2 
into  the  corresponding  instantaneous  projections.  It  is  there- 

'!  . * Blakesley’s  Alternating  Currents  of  Electricity. 

& 2ol 


Fig.  148.  — Rotating  Vector. 


258 


ALTERNATING  CURRENTS 


fore  possible  to  represent  all  the  elements  of  a sinusoidal  func- 
tion : (1)  by  a straight  line  which  rotates  at  a uniform  rate 
around  one  end;  and  (2)  by  the  instantaneous  projections  of 
the  line.  It  is  evident  that  the  motion  which  the  projection  of 
the  end  A of  the  rotating  line  makes  along  the  axis  OY  is  a 
simple  harmonic  motion,  and  that  all  the  theorems  relating  to 
simple  harmonic  motion  may  be  applied  to  these  solutions. 
As  is  ordinarily  done,  the  rotation  of  the  line  will  always  be 
considered  to  be  left-handed ; and  angles  measured  from  right 
to  left  will  be  considered  positive,  while  those  measured  from 
left  to  right  will  be  considered  negative. 

If  two  sinusoidal  voltages  of  the  same  frequency,  but  having 
a phase  difference  6,  act  in  a circuit,  the  corresponding  instan- 
taneous values  are, 

e = V2  E sin  «, 
e'  — V2  E'  sin  («  + #). 


The  total  instantaneous  voltage  acting  in  the  circuit  is  e -f  e' . 
In  Fig.  119,  the  voltage  E is  represented  by  the  line  OA , 

and  the  voltage  E'  by 
the  line  OA' . Oa  and 
Oa'  are  the  instanta- 
neous values  of  the 
voltage  for  the  angular 
positions  shown.  The 
total  instantaneous 
voltage  in  the  circuit 
which  corresponds  to 
the  angular  position 
shown  is  equal  to  Oa 
+ Oa',  or  Oa".  It  is 
readily  shown  that  Oa"  is  the  projection  of  the  diagonal  of  the 
parallelogram  constructed  upon  OA  and  OA' . This  is  true  for 
all  angular  positions,  since  the  sum  of  the  projections  of  the 
lines  OA  and  OA'  must  be  equal  to  the  sum  of  the  projections 
of  the  lines  OA  and  AA" , which  in  turn  is  equal  by  construc- 
tion to  the  projection  of  the  diagonal  OA". 

The  length  of  the  line  OA"  therefore  proportionally  repre- 
sents the  magnitude  of  the  effective  or  maximum  total  voltage 
in  the  circuit,  and  its  position  relative  to  that  of  OA  and  OA! 


SOLUTION  OF  CIRCUITS 


259 


represents  the  relative  phase  position  of  the  total  voltage.  If, 
instead  of  two  voltages  acting  in  a circuit,  there  are  three  or 
more,  as  OA,  OA',  OA ",  OA'",  and  OA "",  in  Fig.  150,  the 
same  construction  is  used.  Thus,  completing  the  parallelo- 
gram for  OA  and  OA',  their  resultant  OAx  is  found.  Com- 
pleting the  parallelogram  for  OAx  and  OA",  their  resultant 
OA2  is  found,  and  again  with  this  and  OA"'  the  resultant  OAz 
is  obtained ; finally  OA4,  the  final  resultant,  is  obtained  by 
combining  OA3  with  OA"" . The  figure  shows  that  it  is  un- 
necessary to  complete  all  the  parallelograms.  It  is  only  neces- 


Fig.  150.  — Resultant  of  Three  or  more  Vectors. 


sary  to  draw  the  lines  OA,  AAV  AXA2,  A2A3,  ^-3^4  respectively 
parallel  and  equal  in  length  to  the  lines  OA,  OA',  OA"  OA'", 
and  OA"",  and  the  line  drawn  from  0 to  the  end  of  the  last 
line  laid  off  gives  the  phase-position  and  magnitude  of  the  total 
voltage  in  the  circuit,  regardless  of  the  number  of  the  compo- 
nents from  which  it  is  derived  (Fig.  151).  The  composition  of 
voltages  is  therefore  exactly  analogous  to  the  composition  of  velo- 
cities or  of  forces.  ^4.s  in  the  case  of  velocities  or  forces,  the  re- 
sultant of  any  number  of  voltages  may  be  determined  by  this  method. 

The  resultant  of  two  sinusoidal  alternating  currents  which 


260 


ALTERNATING  CURRENTS 


flow  in  a divided  circuit  may  be  graphically  determined  in  the 
same  manner.  In  Fig.  149,  let  OA  and  OA!  be  the  currents 
in  the  two  reactive  branches  of  a divided  circuit.  The  two 
partial  currents  differ  from  each  other  in  phase  by  an  angle  6. 
The  instantaneous  values  of  the  currents  are  represented  by 
the  instantaneous  projections  of  the  lines  as  they  revolve 
around  the  point  0.  At  each  instant,  the  total  current  in  the 
main  circuit  is  equal  to  the  sum  of  the  instantaneous  partial 
currents,  or  to  i + i' . Consequently,  the  magnitude  of  the 
effective  or  maximum  value  of  the  current  in  the  main  circuit 
is  proportionally  represented  by  the  length  of  the  line  OA", 
and  the  angular  direction  of  OA"  gives  the  angular  relation  of 


the  phase  of  the  total  current  to  the  phases  of  the  partial  cur- 
rents. When  the  divided  circuit  contains  more  than  two 
branches,  the  same  method  may  be  extended  as  already  ex- 
plained for  the  composition  of  voltages. 

For  convenience  in  using  the  graphical  methods  for  solving 
alternating-current  problems,  it  is  well  to  distinguish  between 
two  different  diagrams.  The  first  diagram  represents  the  mag- 
nitude and  relative  phase  positions  of  the  voltages  or  currents 
by  means  of  lines  radiating  from  a point.  This  may  be  called 
the  Phase  diagram.  It  is  analogous  to  the  force  diagram  of 
graphical  statics.  The  other  diagram  is  the  polygon  formed  by 
laying  off  lines  equal  and  parallel  to  the  lines  in  the  phase 
diagram.  This  may  be  called  the  Vector  diagram  or  polygon. 
It  is  analogous  to  the  funicular  polygon  of  graphical  statics. 


SOLUTION  OF  CIRCUITS 


261 


Figures  150  (full  lines  only)  and  151  are  respectively  phase  and 
vector  diagrams  for  representing  five  voltages  or  currents. 
The  resultant  voltage  is  represented  in  magnitude  and  phase 
by  the  line  OA4.  If  the  closing  line  OAi  of  the  vector  polygon 
is  inserted  in  the  phase  diagram  by  drawing  from  0 a line  in 
the  direction  obtained  by  following  round  the  vector  diagram 
against  the  direction  in  which  the  lines  were  drawn  (that  is, 
from  0 to  A4),  the  line  so  inserted  evidently  represents  the 
resultant  of  the  component  voltages  or  currents.  If  the  line 
he  drawn  from  0 in  the  opposite  direction,  it  represents  a 
balancing  voltage  or  current. 

These  simple  propositions,  which  so  evidently  come  from  the 
ordinary  graphical  mechanics  (statics  and  dynamics),  give  all 
the  foundation  that  is  usually  necessary  for  the  solution  of 
problems  relating  to  the  flow  of  current  in  simple  and  compound 
circuits  containing  definite  resistances,  inductances,  and  capac- 
ities in  their  different  parts.  For  solutions  of  complicated 
problems  the  analytical  method  using  complex  quantities  is 
usually  preferable  to  the  graphical.  The  graphical  method 
has  the  advantage  of  showing  directly  to  the  eye  the  relative 
phases  of  the  voltages  or  currents  in  different  parts  of  the 
circuit,  but  a drawing  board  is  not  always  so  convenient  a 
medium  as  a tablet  of  computing  paper  and  the  sharpness  of 
the  lag  angle  sometimes  to  be  dealt  with  may  cause  indefinite 
intersections  and  inaccuracy  in  graphical  solutions.  The  graph- 
ical solutions  have  the  same  limitations  in  regard  to  alternating 
currents  or  voltages  which  are  not  sinusoidal  as  have  the  ana- 
lytical methods ; and  where  the  wave  form  is  not  sinusoidal, 
only  an  approximation  can  be  arrived  at  by  judiciously  cor- 
recting the  results  shown  by  the  diagrams  based  on  equivalent 
sine  functions,  unless  the  component  sinusoids  are  used. 

80.  Classification  of  Circuits.  — The  problems  relating  to  alter- 
nating-current circuits  which  can  be  solved  by  graphical  meth- 
ods may  be  divided  into  three  classes : (1)  where  the  cur- 
rent flows  through  all  parts  of  the  circuit  in  series ; (2)  where 
the  same  voltage  is  impressed  upon  all  parts  of  the  circuit 
(parallel  circuits)  ; and  (3)  where  the  first  and  second  classes 
are  combined.  Solutions  in  the  third  class  are  effected  by 
combining  partial  solutions  of  the  first  and  second  classes. 
Such  problems  can  also  readily  be  solved  by  the  use  of  the  com- 


262 


ALTERNATING  CURRENTS 


plex  quantity.  The  articles  immediately  following  give  a num- 
ber of  illustrative  problems  which  are  solved  by  both  methods. 

In  more  complicated  systems,  it  is  frequently  desirable  to 
use  the  relations,  — usually  termed  Kirclioff’s  Laws,  — that  the 
algebraic  sum  of  all  the  currents  at  any  junction  point  in  a 
circuit , or  network  of  circuits , is  zero;  and  that  the  algebraic 
sum  of  the  voltages  in  a closed  circuit , or  any  closed  loop  of  a 
network  of  circuits , is  zero.  In  the  first  case,  the  currents 
entering  the  point  under  consideration  are  usually  termed  posi- 
tive and  those  leaving  it,  negative.  In  the  second  case,  it  is 
usual  to  follow  around  the  circuit  in  a clockwise  direction, 
and  call  those  voltages,  in  the  several  parts  or  sections  of  the 
complete  circuit,  positive  which  are  considered  to  increase  from 
one  terminal  of  a section  to  the  next,  and  those  negative  which 
are  considered  to  decrease  from  one  terminal  to  the  next. 
These  relations  are  in  accordance  with  the  simple  laws  of  elec- 
tricity and  in  the  form  above  are  scarcely  worthy  of  statement 
as  separate  laws,  but  are  rather  convenient  expressions  of  Ohm's 
law.  The  relations  should  be  carefully  borne  in  mind  as  here 
stated,  for  the  purpose  of  affording  systematic  direction  to  the 
solution  of  complex  problems. 

81.  Series  Circuits.  — First  Class.  — Suppose  a circuit  is  given 
which  has  a certain  resistance,  self-inductance,  and  capacity,  and 
it  is  desired  to  know  what  impressed  voltage  with  a frequency 
f is  required  to  pass  through  it  a certain  current  I.  In  this  case 
the  impressed  voltage  is  made  up  of  two  components:  (1)  the 
voltage  equal  to  the  IR  drop  caused  by  the  current  flowing 
through  the  resistance  of  the  circuit,  which  we  call  the  active 
voltage ; (2)  the  voltage  required  to  balance  or  overcome  the 
reactive  counter-voltage.  The  reactive  voltage  is  90°  in  phase 
in  advance  of  or  behind  the  active  voltage,  depending  upon 
whether  the  reactance  lias  the  effect  of  capacity  or  of  induct- 
ance, and  the  phase  diagram  which  shows  the  relative  phases 
of  the  voltages  in  the  circuit  is  therefore  like  that  shown  in 
Fig.  152.  The  active  voltage  OA'  is  equal  to  IR.  and  the 
reactive  voltage  OA"  is  equal  to  2 irfLI,  this  representing  a 
circuit  with  inductive  reactance.  The  reactive  component  of 
the  impressed  voltage  is  required  to  balance  2 irfLl \ and  is 
therefore  equal  to  and  opposite  to  OA".  An  arrowhead  is 
therefore  placed  on  OA'  to  show  that  in  the  vector  pol}'gon  its 


SOLUTION  OF  CIRCUITS 


263 


direction  must  be  taken  from  A"  to  0,  instead  of  from  0 out- 
wards, as  is  done  with  the  other  lines.  The  vector  polygon  is 
therefore  given  by  drawing  01A1 
equal  and  parallel  to  OA',  A1A2 
equal  and  parallel  to  A"0 , and  clos- 
ing the  polygon  by  the  line  0XA2 
(Fig.  152).  The  line  OxA2  taken  in 
the  direction  from  0X  to  A2  repre- 
sents the  magnitude  and  relative 
phase  of  the  impressed  voltage. 

When  inserted  in  the  phase  dia- 
gram, it  is  the  line  OA'" . The 
angle  d,  by  which  the  current  lags 
behind  the  impressed  voltage,  is  the 
angle  A101A2. 

If  a number  of  inductive  circuits 
are  connected  in  series,  the  line  in 
the  phase  diagram  which  represents 
the  reactive  voltage  has  a length 
equal  to  2 i rfI(L1+  L2  + Ls+  etc.), 
and  the  line  which  represents  the  active  voltage  has  a length 
equal  to  I(R1  -4-  R2  + R3  + etc.),  where  Lv  Lv  L3 , Rv  R2,  R3 , 
etc.,  are  the  self-inductances  and  resistances  of  the  different 
parts  of  the  circuit.  If  the  circuit  is  non-reactive,  the  phase 
diagram  and  vector  polygon  each  become  a single  horizontal 
line  equal  in  length  to  IR,  while  if  the  inductive  circuit  con- 
tains no  resistance,  the  diagrams  each  become  a single  vertical 
line  which  is  equal  in  length  to  2 71 fLI. 

Dividing  the  lengths  of  the  sides  of  a vector  polygon  of  vol- 
tages by  the  value  of  I in  the  circuit  gives  the  impedances  of 
the  several  parts  of  the  series  circuit.  A vector  polygon  of  the 
voltages  impressed  on  the  parts  of  a series  circuit  and  a polygon 
of  the  impedances  of  the  corresponding  parts  may  evidently  be 
converted,  one  into  the  other,  by  a simple  change  of  scale.  It 
is  to  be  remembered  that  in  a series  circuit  the  current  has  the 
same  phase,  and  is  equal  at  any  given  instant,  in  all  parts  of 
the  circuit ; but  the  phase,  with  reference  to  the  current,  of  the 
voltage  impressed  at  the  terminals  of  different  parts  of  the  cir- 
cuit depends  for  each  part  wholly  upon  the  relation  between 
the  individual  resistance  and  reactance  of  the  part. 


Diagrams  of  Self-inductive 
Circuit. 


264 


ALTERNATING  CURRENTS 


Examples.  — In  the  following  examples  it  is  desired  to  find 
for  each  of  the  given  circuits  : (1)  the  impedance  of  the  cir- 
cuit ; (2)  the  angle  by  which  the  current  lags  behind  the 
impressed  voltage ; (3)  the  current  which  flows  through  the 
circuit  when  the  impressed  voltage  is  100  volts ; (4)  the  im- 
pressed voltage  which  is  required  to  pass  10  amperes  through 
the  circuit.  The  frequency  in  each  case  is  taken  just  under 
127-|  periods  per  second,  whence  2 irf  is  equal  to  800,  and 
current  and  voltage  are  assumed  to  be  sinusoidal. 


Circuits  containing  Resistance  and  Seif-inductance 


a.  The  circuit  is  non-reactive  and  has  a resistance  of  10  ohms. 
The  phase  and  vector  diagrams  for  the  first  solution  each  con- 
sist of  a horizontal  line  10  units  in  length. 


R=io 


| the 


The  impedance  of  the  circuit  is  10  ohms,  and 


current  which  flows  when 


voltage 


of 


100  volts  is  impressed  on  the  circuit  is  10 
O R = 1°  A amperes.  The  diagrams  for  the  fourth  solu- 

oi  IR  ioo  tion  eacp  coi:isist  of  a horizontal  line  IR(  = 

Fig. 153. — Solutiono.  ...  . . . 

100)  units  m length,  and  the  impressed  vol- 
tage required  to  pass  10  amperes  through  the  circuit  is  100  volts. 
The  angle  6 is  zero.  The  complex  expression  for  the  imped- 
ance of  the  circuit  is 


Z = r +jx  = 10  4-/0. 

Therefore, 

(1)  Z = v/102  -|-  02  = 10  ohms,  and  (2)  6 = tan-1^  = 0°. 
When  the  impressed  voltage  is  100, 


and  (3)  I—  = 10  amperes,  in  phase  with  the  vol 
When  the  current  flowing  through  the  circuit  is  10  anq 


E—JZ=  Ir  4-/70, 
or  R = 1 0 R 

and  (4)  E = IZ  = 100  volts,  in  phase  with  the  current. 

b.  The  circuit  consists  of  an  inductive  coil  of  10  ohms  re- 
sistance and  0.01  henry  self-inductance.  The  phase  and  vector 


SOLUTION  OF  CIRCUITS 


265 


diagrams  for  the  solutions  are  shown  in  Fig.  154.  The  imped- 
ance is  shown  by  the  closing  line  of  the  impedance  diagram  to 
be  12.8  ohms,  whence  it  is  seen  that  7.81  amperes  will  flow 
through  the  circuit  when  the  impressed  voltage  is  100  volts. 


R = 10  L=  .01 


Fig.  154.  — Solution  b. 


The  fourth  solution  shows  that  the  impressed  voltage  required 
to  pass  10  amperes  through  the  circuit  is  128  volts.  The 
angle  6 is  38°  40'.  The  complex  expression  for  the  impedance 
in  the  circuit  is 

Z — r +jx  — 10  ,+y  8 


and  (1)  Z=  VlO2  + 82  = 12.8  ohms. 

(2)  6 = tan-1  =f$8°  40'. 

When  the  impressed  voltage  is  100  volts, 


and 


Y Er  .Ex  1000  .800 

J=-®y=22-^=i6r^i6i’ 

(3)  /=  100  x — — = 7.81  amperes,  lagging  behind  the 

12.8 


voltage. 

The  two  terms  in  the  right-hand  member  of  the  complex 
expression  are  respectively  the  two  rectangular  components, 
Ig  and  Ib , of  the  current. 


266 


ALTERNATING  CURRENTS 


When  the  current  flowing  through  the  circuit  is  10  amperes, 
E = IZ  = Ir  +jlx  = 100  +/  80, 

and  (4)  E — IZ  = 10  x 12.8  = 128  volts,  leading  the  current. 

The  two  terms  in  the  right-hand  member  of  the  complex  ex- 
pression, namely  Ir  and  lx , are  respectively  the  active  and 
reactive  components  Er  and  Ex  of  the  impressed  voltage. 

c.  The  circuit  consists  of  a non-inductive  coil  of  5 ohms  in 
series  with  an  inductive  coil  of  5 ohms  and  0.01  henry.  The 

total  phase  and  vector 
R = 5 01  diagrams  for  this  are  simi- 

lar  to  those  in  example  b. 
The  vector  diagram  may 
be  laid  off  as  shown  in 
Fig.  155.  This  shows 
that  the  voltage  OxA3  im- 
pressed upon  the  circuit 
as  a whole  is  not  equal 
to  the  algebraic  sum  of 
the  voltages  OxA2  and 
A2A3  measured  between 
the  terminals  of  the  parts  of  the  circuit,  but  it  is  still  equal  to 
their  vector  sum. 

The  solution  by  means  of  complex  expressions  is  as  follows : 
Z1=  5 +/  8 tan  6 = = . 8.  6 = 38°  40' 


ta" 

27T/L 

= 8 

. ///' 

R=5  * 

Z^A38°40 

R = 5 A> 


Fig.  155.  — Solution  c. 


5 +/  0 


Z = Vr2+:r2=VT02-t-82  = 12.8  ohms. 


Z =10+/ 8 

When  100  volts  are  impressed  on  the  circuit, 

I=EY=^-j- lift 

and  (3)  1=  7.81  amperes,  lagging  behind  the  voltage. 

When  10  amperes  flow  through  the  circuit, 

E=IZ=  100+/ 80, 


and  (4)  E = 10  x 12.8  = 128  volts,  leading  the  current. 

d.  The  circuit  comprises  only  inductance  of  0.01  henry.  The 
phase  and  vector  diagrams  for  the  first  solution  each  consist  of 
a vertical  line  2 7 rf.L(=  8)  units  in  length.  The  impedance  of 
the  circuit  is  8 ohms,  and  the  current  which  flows  through  the 


SOLUTION  OF  CIRCUITS 


267 


u 

■J 

27T/L 

8 

'iTTj'i  I 

= 8 

*-^0=90° 

= 80 

A„ 

Ai 


«o 


circuit  under  an  impressed  voltage  of  100  volts  is  12.5  amperes. 
The  diagrams  for  the  fourth  solution  L=  oi 

each  consist  of  a vertical  line  2 irfLI  r^^TSinrirVi 

(=80)  units  in  length,  and  the  im- 
pressed voltage  required  to  pass  10 
amperes  through  the  circuit  is  80 
volts.  The  angle  6 is  90°,  and  the 
current  therefore  lags  90°  behind  the 

impressed  voltage.  hxe=90°  b\0=9o° 

(1)  Z=0  +y  8,  Z=  8,  a' 

(2)  6 = tan-1 1 = 9Q°. 

When  a voltage  of  100  is  impressed  on  the  circuit, 

Y 100  .100 

Z J 8 ’ 

and  (3)  -§-=  12.5  amperes,  lagging  behind  the  voltage. 

When  10  amperes  are  flowing  through  the  circuit, 

^=ioz=y  so, 

and  (4)  _Z?=  80  volts,  leading  the  current. 


O, 


A' 


O, 


Fig.  156.  — Solution  d. 


The  effect  of  a condenser  placed 
in  series  in  a circuit  may  be  shown 
by  diagrams  which  are  very  similar 
to  those  relating  to  inductive  circuits. 
The  charging  current  of  the  con- 
denser has  such  a phase  position  and 
magnitude  that  its  effect  on  the  total 
current  flowing  in  the  circuit  is  the 
same  as  the  effect  of  a voltage  which 


is  equal  to 


I 


and  is  90°  in  ad- 


2t jfO 

vance  of  the  circuit  current.  This 
may  be  called  the  Condenser  vol- 
tage.* The  phase  diagram  is  like 
A2  that  shown  in  Fig.  157.  The  vol- 
Phase  and  Vector  Dia-  tage  impressed  on  the  circuit  when 
current  I flows  must  consist  of  two 
components : (1)  the  voltage  required  to  pass  the  current 


Fig.  157 

grams  of  a Capacity  Circuit. 


* Art.  54. 


2(58 


ALTERNATING  CURRENTS 


through  the  resistance  of  the  circuit:  (2).  the  voltage 

required  to  balance  the  condenser  voltage.  The  active 
voltage  OA  in  Fig.  157  is  equal  to  IR,  and  the  condenser 


voltage  OA"  is  equal  to 
active  voltage 


and  is  90°  in  advance  of  the 


2 77/(7 

The  capacity  component  of  the  impressed 


voltage  is  required  to  balance 


I 


2 t rfC' 


and  is  therefore  equal 


and  opposite  to  OA".  An  arrowhead  is  therefore  placed  on 
OA"  to  show  that  in  the  vector  polygon  its  direction  must  be 
taken  from  A"  to  0,  instead  of  from  0 outwards.  The  vector 
polygon  is  then  as  shown  in  Fig.  157  (compare  with  Fig.  152). 

Examples.  — The  following  examples  are  to  be  solved  for  the 
quantities  as  before,  the  value  of  2 nf  being  taken  as  800  and 
voltage  and  current  being  assumed  to  be  sinusoidal. 

Circuits  containing  Resistance  and  Capacity 

e.  The  circuit  contains  simply  a condenser  having  a capacity 
of  100  microfarads  (=  0.000100  farad).  The  phase  and  vector 
diagrams  for  the  first  solution  each  consist  of  a vertical  line 


- — 12.5)  units  in  length.  The  impedance  of  the  circuit 

is  12.5  ohms,  and  the  current  which  flows  through  the  circuit, 

when  100  volts  is  impressed  on  it,  is 
8 amperes.  The  diagrams  for  the 
fourth  solution  each  consist  of  a ver- 


C=ioo 


A' 


O,  A' 

‘ *“T“ 

0>-9O 


o, 


tical  line 


I 


27 rfc 
=12.5 


12.6 


1 

27 rfc 
=126 


A, 


0/1-90° 


irfC 


( = 125)  units  in 


126 


A, 


Fig.  158.  — Solution  e. 


length,  and  the  impressed  voltage  re- 
quired to  pass  10  amperes  through 
the  circuit  is  125  volts.  The  angle 
6 is  — 90°;  that  is,  the  current  is  90° 
in  advance  of  the  impressed  voltage. 
The  lines  composing  the  diagrams 
for  this  example  are  drawn  in  a direction  which  is  exactly 
opposite  to  that  of  the  lines  in  the  diagrams  of  the  example  d. 

(1)  Z = 0 —j  12.5,  Z — 12.5  ohms. 

(2)  6 = tan"1  - = - 90°. 


SOLUTION  OF  CIRCUITS 


269 


When  the  impressed  voltage  is  100, 


and 


T 100  .100  . q 

1 — = 7 = 78, 

£ J 12.5  J 

(3)  I—  ~~  = 8 amperes,  leading  the  voltage. 

±A,  u 


When  the  current  flowing  through  the  circuit  is  10  amperes, 
E = 10  Z = - j 125, 

and  (4)  E = 10  x 12.5  = 125  volts,  lagging  behind  the  current. 
/.  The  circuit 


R =io 


C '=  ioo 


2tt/C 
=12. 5 


R =10 


A' 


contains  a resist- 
ance of  10  ohms  and 
a capacity  of  100  A 
microfarads.  The 
phase  and  vector 
diagrams  are  shown 
in  Fig.  159.  The 
impedance  of  the 
circuit  is  16  ohms 
and  the  current  ° 
which  flows  under 
an  impressed  vol- 
tage of  100  volts  is  6.25  amperes.  The  impressed  voltage  re- 
quired to  cause  10  amperes  to  flow  through  the  circuit  is  160 
volts.  The  angle  6 is  — 51°  20' 

(1)  Z = 10  — j 12.5,  Z = 16  ohms, 

(2)  S = tan"1  - ^ = - 51°  20'. 

v 7 10 

For  100  volts  impressed, 


Fig.  159.  — Solution  /. 


7=100  Y = — + j 1250 


256 


256  ’ 


and 


100 


(3)  I = — = 6.25  amperes,  leading  the  voltage. 
Zj 


For  10  amperes  flowing, 

E=  10  Z=  100  -j  125, 

and  (4)  E = 10  x 16  = 160  volts,  lagging  behind  the  current. 


270 


ALTERNATING  CURRENTS 


Circuits  containing  Self -inductance  and  Capacity 


C=ioo  L = .oi 

=^nmnri 


o-- 


♦ A, 


2jt/C 

=12.5 


2tt/L 
= 8 


/-90° 


4.5 


Fig.  160.  — Solution  g. 


g.  The  circuit  consists  of  a con- 
denser of  100  microfarads  capacity 
in  series  with  a 0.01  henry  induct- 
ance, the  resistance  being  negligi- 
ble. The  diagrams  are  as  shown 
in  Fig.  160.  The  impedance  of  the 
circuit  is  4.5  ohms.  The  current 
which  flows  when  the  impressed 
voltage  is  100  volts  is  22.2  amperes, 
at  which  time  the  voltage  measured 
between  the  terminals  of  the  con- 
denser is  277.5  volts  and  that  meas- 
ured between  the  terminals  of  the 
inductance  is  177.6  volts.  The  im- 
pressed voltage  required  to  pass  10 
amperes  through  the  circuit  is  45 


90° 


volts.  The  angle  0 is 

The  complex  expression  for  impedance  is 

Z\  = 0 +j  8.0 
Z,  = 0 -j  12.5 


and 


Z = 0-j  4.5 
(1)  Z=  4.5  ohms. 


(2)  The  angle  of  lag  is  0 — tan" 


T5 

0 


= -90° 


(3)  When  the  impressed  voltage  is  100, 

T_100_  .100 

Z J 4. 5’ 

I — = 22.2  amperes,  leading  the  voltage. 

Z 

(4)  When  10  amperes  flow,  the  impressed  voltage  is 


E — IZ  = — j 10  x 4.5  = — j 45, 

E = 1Z  = 45  volts,  lagging  behind  the  current. 

The  voltage  measured  between  the  terminals  of  the  inductive 
coil  when  10  amperes  flow  is 

Ex  = 1ZX  = j 80. 

Ex  = 80  volts,  leading  the  current, 


SOLUTION  OF  CIRCUITS 


271 


L = .016 

r^nnr- 


C = 125 


A, 


and  the  voltage  measured  between  the  terminals  of  the  con- 
denser is  ^7 

= IZ2=  -3  125, 

= 125  volts,  behind  the  current. 

Under  a current  flow  of  22.2  amperes,  the  voltage  measured 
between  the  terminals  of  the  condenser  is  22.2  x 12.5  = 277.5 
volts,  and  the  voltage  measured  between  the  terminals  of 
the  inductive  coil  is  22.2  x 8 = 177.6 
volts  as  pointed  out  in  the  first 
paragraph. 

h.  The  circuit  comprises  a capacity 
of  125  microfarads  in  series  with  an 
inductance  of  0.015  henry,  the  resist-  a'x 
ance  being  negligible.  The  diagrams 
of  Fig.  161  show  that  impedance  of  2tt/c 
the  circuit  is  2 ohms  and  the  current  =1° 
which  flows  under  100  impressed  volts 
is  therefore  50  amperes,  at  which  time 
the  voltage  measured  between  the  ter- 
minals of  the  condenser  is  500  volts 
and  between  the  terminals  of  the  in- 
ductance coil  is  600  volts.  The  im- 
pressed voltage  required  to  pass  10 
amperes  through  the  circuit  is  20  volts. 

With  this  current  flowing,  the  voltage  A'-L 
measured  between  the  terminals  of  the 
condenser  is  100  volts  and  between  the  terminals  of  the  in- 
ductance is  120  volts.  The  angle  6 is  90°. 

Solving  by  the  complex  expressions  gives, 

Z1  = 0 + j 12  Z = Vo2  + 22  = 2 ohms 

a .‘in  o 

0 = 90°. 


o ■■ 


1 

1 

k-~ 

i 

<M 

tH 

01 — 

!*. 



4'- 

<4 

* 

.0=90° 

X 

O, 


2tt/U 

=12 


Fig.  161.  Solution  h. 


tan  0 = - = oo 
0 


5 = o m io 

z = 0+3  2 

When  100  volts  are  impressed  upon  the  circuit,  the  current  is 

T U .100  . „ 

Z 2 

and  I — 50  amperes,  lagging  behind  the  voltage. 

The  voltage  measured  between  the  terminals  of  the  condenser 
at  this  time  is 

and 


E2  = IZ2  = 50  x — j 10  = — j 500, 
E„  = 500  volts,  behind  the  current. 


272 


ALTERNATING  CURRENTS 


The  voltage  measured  between  the  terminals  of  the  inductance 
coil  is 

Ex  = 1ZX  = 50  x/  12  = j 600 
and  Ex  — 600  volts,  ahead  of  the  current. 


L =.0156 


a" 


1 

27T/C 

=12.5 


•>  f 

o-- 

, V 


C = ioo 


Ai 

'FT*' 


.i.l.t. 
0,  A, 


27T/L 
= 12.5 


A'.. 

Fig.  162.  — Solution  i. 


It  will  be  observed  that  the  500 
volts  measured  between  the  con- 
denser terminals  and  the  600  volts 
measured  between  the  coil  terminals 
are  in  exact  opposition,  and  the  vol- 
tage measured  between  the  termi- 
nals of  the  circuit  made  up  of  the 
condenser  and  coil  in  series  is  the 
difference  of  the  two  or  100  volts. 
This  combination  gives  a large  rise 
of  voltage  at  the  condenser  and  at 
the  coil  caused  by  the  interaction 
of  the  two  through  their  mutual 
transfer  of  energy. 

When  current  flowing  through 
the  circuit  is  10  amperes,  the  vol- 
tage measured  between  the  circuit 
terminals  is 


E=IZ  = j 20 

and  E = 20  volts,  ahead  of  the  current. 

Under  these  circumstances 


E2  = IZ.2=  - j 100 

and  E2  = 100  volts,  behind  the  current. 

Ex  = IZX  = +j  120 
Ex  — 120  volts,  ahead  of  the  current. 
i.  The  circuit  comprises  a capacity  of  100  microfarads  in 
series  with  an  inductance  of  0.0156  henry,  resistance  being 
negligible.  The  phase  and  vector  diagrams  are  shown  in  Fig. 


162.  In  this  case 


2 irfC 


= 2 7i fL,  OxAx  is  equal  and  opposite 


to  AxA2,  and  the  impedance  of  the  circuit  is  zero. 


Zx  = 0 +j  12.5  Z=  0 

Z2  = 0 — j 12.5  That  is,  this  acts  like  a 

Z = 0 short  circuit. 


SOLUTION  OF  CIRCUITS 


273 


If  a current  of  10  amperes  is  caused  to  flow  through  this 
arrangement  of  capacity  and  inductance  in  series,  the  voltage 
measured  between  the  terminals  is  zero ; but  the  voltage  meas- 
ured between  the  terminals  of  the  condenser  is 


and 


E2  = IZ2  = - j 125 

E2  = 125  volts,  behind  the  current ; 


and  the  voltage  measured  between  the  terminals  of  the  induct- 
ance coil  is  --tor 

Ex  = IZi  =j  125 

and  Ex  = 125  volts,  ahead  of  the  current. 

Circuits  containing  Resistance , Self- Inductance,  and  Capacity 


R — lo  L=.oi 


C = ioo 


j.  The  circuit 
consists  of  10  ohms 
plain  resistance, 

100  microfarads, 
and  .01  henry  in 
series  relation. 

The  impedance  of 
the  circuit  is  shown 
by  Fig.  163,  to  be 
10.96  ohms.  The 
current  which 
flows  through  the 
circuit  when  100 
volts  are  impressed 
at  its  terminals  is 
9.12  amperes.  The 
voltage  required  to 
pass  10  amperes 
through  the  circuit 
is  109.6  volts.  This 
is  the  vector  sum 
of  125  volts  and 
128  volts,  which  are 
the  voltages  meas- 
ured respectively 
between  the  condenser  terminals  and  the  remainder  of  the  cir- 
cuit. The  angle  9 is  — 24°  14'. 


A'"" 

l 

2jt/C 
= 12.5 

R=lo  A' 

27T/L 
= 8 

A" 

R =10,  L=  01 

Fig.  163.  — Solution  j. 


274 


ALTERNATING  CURRENTS 


Solved  by  complex  quantities,  there  results 

zx  = 10  +j  0 

Z2  = 0 -\-j  8 

3a=0-/12.5 
3 = 10-/ 4.5 

Here  the  prefix  of  the  reactance  term  in  the  expression  for 
3 is  — /,  hence  the  angle  6 is  negative. 

When  100  volts  are  impressed  on  the  circuit, 

I_E_FY-  1000  • 450 

z J (10.96)2+';(10.96)2 

and  I—  = 9.12  amperes,  leading  the  voltage. 

Under  these  conditions,  the  voltage  measured  across  the  ter- 
minals of  the  condenser  is 

Ez  = IZz  = — / 9.12  x 12.5 
and  ^3=  114  volts,  behind  the  current. 

The  voltage  measured  across  the  remainder  of  the  circuit  is 
E1+2  = KZX  + Z3)  = 9.12  (10  +/  8) 

and  E1+2  = 4-  72.9B2  = 116.8  volts,  ahead  of  the  current. 

When  10  amperes  is  caused  to  flow  through  the  circuit, 

J=  73=  100-/45 

and  E = 109.6  volts,  behind  the  current. 

The  values  of  E3  and  Ex+2  under  these  circumstances  may  be 
computed  in  the  same  manner. 

k The  circuit  comprises  10  ohms  plain  resistance,  150  micro- 
farads, and  01042  henry  in  series.  The  diagrams  in  Fig  164 
show  that  the  impedance  of  the  circuit  is  10  ohms  One  hun- 
dred volts  is  therefore  the  impressed  voltage  that  gives  a cur- 
rent of  10  amperes.  When  10  amperes  flow  in  the  circuit,  the 
voltage  measured  between  the  terminals  of  the  condenser  is 
83.8  volts,  and  that  measured  between  the  terminals  of  the 
remainder  of  the  circuit  is  180  volts.  The  angle  6 is  zero. 

Zx  = 10  +/  0 3=10  ohms. 

Z2  = 0 + j 8. 33  tan  0 _ q _ qo_ 

33  = 0 — j 8.33  10 

Z =10+/0 


Z = VlO2  + 4.52  = 10.96  ohms, 
tan  e = - 6 = - 24°  14'. 


SOLUTION  OF  CIRCUITS 


275 


When  100  volts  are  impressed  on  this  circuit, 


100 

10 


= 10 


and  I—  10  amperes,  in  phase  with  the  voltage. 

The  voltage  measured  between  the  terminals  of  the  condenser 
(assuming  it  to  be  resistanceless)  is 

j?3  = -yiO  X 8.38  = -/  83.3 
and  Eq  = 83.3  volts,  behind  the  current. 

The  voltage  measured  between  the  terminals  of  the  inductance 
coil  (assuming  it  to  be  resistanceless)  is 

E2  = +j  10  x 8.33  = +j  83.3 
and  _Z?2  = 83.3  volts,  ahead  of  the  current. 


i 

27T/C 
= 8.33 


O 


R =10 


A' 


2 tt/L 
=8.33 


These  two  reactive  R = 10  L = .01042 
components  of  vol- 
tage  are  equal  and 
opposite  to  each 
other,  and  the  vol- 
tage measured 
across  the  circuit  is 
the  same  as  that 
measured  across  the 
plain  resistance  coil, 
namely,  100  volts  in 
phase  with  the  cur- 
rent. 

82.  Conclusions  in 
Regard  to  Series  Cir- 
cuits. — The  eleven 
examples  thus  given 
cover  every  funda- 
mental arrangement 
of  series  circuits 
which  may  occur. 

An  examination  of 
the  diagrams  makes 
the  following  state- 


C = 150 


R =10,  L = .0104 


83.3 


t 

8.33 


10 


o, 


Fig.  164.  — Solution  k. 


ments  evident  for 
circuits  in  which  are  sinusoidal  voltages  and  currents : 


8.33 


R=10,  C =150 


83.3 


276 


ALTERNATING  CURRENTS 


1.  When  non-inductive  circuits  are  connected  in  series,  the 
total  impressed  voltage  equals  the  sum  of  the  voltages  measured 
between  the  terminals  of  the  individual  parts,  and  the  total 
resistance  of  the  circuit  is  equal  to  the  sum  of  the  resistances 
of  the  individual  parts. 

2.  When  inductive  circuits  of  equal  time  constants  are  con- 
nected in  series,  the  total  impressed  voltage  equals  the  sum  of 
the  voltages  measured  between  the  terminals  of  the  individual 
parts,  and  the  total  impedance  of  the  circuit  is  equal  to  the 
sum  of  the  individual  impedances. 

3.  When  inductive  and  non-inductive  circuits  are  connected 
in  series  with  each  other,  or  when  inductive  circuits  of  unequal 
time  constants  are  connected  in  series,  the  total  impressed  vol- 
tage equals  the  vector  sum,  which  is  always  less  than  the  alge- 
braic sum,  of  the  voltages  measured  between  the  terminals  of 
the  individual  parts,  and  the  individual  voltages  are  each  less 
than  the  total  impressed  voltage.  The  total  impedance  of  the 
circuit  is  equal  to  the  vector  sum  of  the  individual  impedances, 
each  of  which  is  less  than  the  total. 

4.  When  condensers  are  connected  in  series  by  conductors  of 
7iegligible  resistance , the  total  impressed  voltage  equals  the  sum 
of  the  voltages  measured  across  the  individual  condensers,  and 
the  total  impedance  of  the  circuit  is  equal  to  the  sum  of  the 
impedances  of  the  individual  condensers. 

5.  When  condensers  are  connected  in  series  with  non-hiduct- 
ive  circuits , the  total  impressed  voltage  equals  the  vector  sum, 
which  is  always  less  than  the  algebraic  sum,  of  the  voltages 
measured  between  the  terminals  of  the  individual  parts  of  the 
series,  and  the  individual  voltages  are  each  less  than  the  total 
impressed  voltage.  The  total  impedance  of  the  circuit  is  equal 
to  the  vector  sum  of  the  individual  impedances,  each  of  which 
is  less  than  the  total. 

6.  When  condense?-s  are  connected  in  series  with  inductive 
circuits , the  total  impressed  voltage  equals  the  vector  sum, 
which  is  always  less  than  the  algebraic  sum,  of  the  voltages 
measured  between  the  terminals  of  the  individual  parts  of  the 
series.  Since  the  effects  of  capacity  and  self-inductance  respec- 
tively cause  the  lag  angle  to  become  negative  and  positive,  the 
individual  voltages  may  be  either  greater  OR  less  than  the  total 
impressed  voltage , depending  upon  the  relation  between  the 


SOLUTION  OF  CIRCUITS 


277 


various  resistances,  capacities,  and  inductances  in  the  circuit. 
The  total  impedance  of  the  circuit  is  equal  to  the  vector  sum 
of  the  individual  impedances,  each  of  which  may  be  either 
greater  or  less  than  the  total  impedance. 

The  third,  fifth,  and  sixth  paragraphs  above  make  the  follow- 
ing proposition  evident : When  in  series  circuits  the  angles 
measured  between  the  phase  of  the  current  and  the  phases  of 
the  individual  voltages  measured  between  terminals  of  parts  of 
the  series,  are  all  either  positive  or  negative,  the  total  impressed 
voltage  is  always  greater  than  any  of  the  individual  or  partial 
voltages.  When  the  angles  measured  between  the  phase  of 
the  current  and  the  phases  of  the  partial  voltages  are  in  part 
positive  and  in  part  negative,  some  or  all  of  the  partial  voltages 
may  be  greater  than  the  total  impressed  voltage. 

83.  Parallel  Circuits.  — Second  Class. — The  graphical  and 
analytical  treatment  of  problems  relating  to  parallel  circuits  is 
entirely  analogous  to  that  given  for  series  circuits.  As  the 
simplest  cases  of  parallel  circuits  are  those  in  which  the  same 
voltage  is  impressed  upon  all  the  parts  of  the  circuit,  these  will 
be  treated  first.  In  this  class  the  same  general  operations  are 
used  in  solving  problems  as  in  the  first  class  (series  circuits), 
but  alternating  currents  and  admittances  are  dealt  with  instead 
of  alternating  voltages  and  impedances.  Suppose  a circuit  is 
made  up  of  two  branches  in  parallel,  each  with  a known  resist- 
ance and  reactance,  and  it  is  desired  to  know  what  impressed 
voltage  with  a frequency  f is  required  to  cause  a sinusoidal 
current  I to  flow  through  the  circuit.  In  this  case  the  total 
current  is  made  up  of  two  components,  each  of  which  flows 
through  one  of  the  branches  and  is  inversely  proportional  to 
the  impedance  of  the  branch,  and  the  phase  of  which  has  an 
angular  retardation  with  respect  to  the  impressed  voltage 
which  depends  upon  the  time  constant  of  the  branch.  The 
total  current,  which  is  inversely  proportional  to  the  equivalent 
impedance  of  the  parallel  circuit,  is  equal  in  magnitude  and 
position  to  the  resultant  of  the  branch  currents.  The  con- 
dition is  represented  by  Fig.  165,  in  which  OA!  and  OA"  are 
the  currents  in  the  two  branches  respectively,  O'  and  6"  being 
their  respective  lag  angles.  The  relative  phase  position  of  the 
impressed  voltage  is  taken  on  the  horizontal  line.  Then  the 
resultant  or  total  current  in  the  circuit  is  represented  in  mag- 


278 


ALTERNATING  CURRENTS 


Fig.  165. — Currents  in  Parallel  Branches. 


E 

nitucle  and  phase  by  OA.  OA!  is  equal  to  — , OA"  is  equal  to 

F E ■ Zx 

— , and  OA  is  equal  to  — where  E is  the  voltage  impressed  on 
Z9  Z 

the  branches  and  the  de- 
nominators of  the  frac- 
tions are  the  respective 
impedances  of  the  branches 
and  of  the  total  circuit.  It 
is  therefore  evident  that 
the  reciprocal  of  the  joint 
impedance  Z(=  the  joint 
admittance)  of  a parallel 
circuit  may  be  at  once  derived  from  the  admittances  of  the 
branches,  by  taking  their  vector  sum,  as  is  shown  in  Fig.  166. 
The  joint  admittance  or  the  joint  impedance  of  a branched  circuit 
being  known  for  a particular  frequency,  the  voltage  of  that  fre- 
quency required  to  pass  a 
given  current  through  it,  or 
the  current  flowing  under  a 
given  impressed  voltage  of 
the  same  frequency,  may 
be  at  once  dei’ived.  In 
dealing  with  series  cir- 
cuits, the  phase  of  the  cur- 
rent (or  active  voltage) 
has  been  assumed  to  be 

along  the  horizontal  line,  .Y. x 

or  line  of  reference.  In 
dealing  with  parallel  cir- 
cuits, it  is  more  convenient 
to  assume  the  phase  of  the 
impressed  voltage  (or  conductance)  for  the  reference  phase. 
It  must  always  be  remembered,  however,  that  angles  of  lag  are 
measured  from  lines  representing  current  to  lines  representing 
voltage.  Thus,  in  Figs.  165  and  166  the  angle  6 is  negative 
because  the  current  leads  the  voltage. 

The  resultant  current  is  also  obtained  by  adding  the  com- 
plex expressions  representing  the  branch  currents.  Thus  the 
resultant  of  two  currents  in  parallel  circuits  is 


Fig. — 166.  Vector  Diagram  of  Admittances. 


SOLUTION  OF  CIRCUITS 


279 


1=  Ix  + 12  = El  \ + E Y2  — E(gx  + gf)  — + ^2)* 

In  this  equation  Egx  + Eg2  is  the  active  component  Ig  of  the 
particular  current,  and  Eb1  + Eb2  the  component  Ib  in  quad- 
rature. In  the  same  way  the  combined  admittance  of  parallel 
circuits  is  obtained.  The  addition  is  performed  graphically  in 
Figs.  165  and  166. 

Examples.  — In  the  following  examples  it  is  desired  to  find 
for  each  of  the  given  circuits  : (1)  the  joint  impedance  of 
the  circuit  at  the  given  frequency,  (2)  the  angle  by  which 
the  total  current  lags  behind  the  impressed  voltage,  (3)  the 
current  which  flows  through  the  circuit  when  the  impressed 
voltage  is  100  volts,  (4)  the  impressed  voltage  which  is  re- 
quired to  pass  10  amperes  through  the  circuit.  The  frequency 
is  taken  as  in  examples  of  the  series  circuit  to  be  just  under 
127^-  periods  per  second,  whence  2 7r/is  equal  to  800 ; and  cur- 
rent and  voltage  are  supposed  to  be  sinusoidal. 


R,  - 10 


Circuits  containing  Resistance  and  Self-inductance 

a.  The  circuit  consists  of  two  non-reactive  * branches  in 
parallel,  one  having  20  ohms  resistance  and  the  other  10  ohms 
resistance.  The  phase  diagram  for 
the  solutions,  using  the  admittances 
as  a basis  of  work,  is  two  horizontal 
lines  superposed,  of  lengths  respec- 
tively .05  and  .10  unit.  The  vector 
diagram  is  obtained  by  drawing 
these  consecutively,  and  the  equiva- 
lent admittance  of  the  circuit  is  OA2 
in  Fig.  167  and  is  equal  to  .15  mho. 

The  joint  impedance  of  the  cir- 
cuit is  therefore  6.67  ohm.  The  current  flowing  under  100 
volts  impressed  is  therefore  15  amperes,  and  the  voltage  re- 
quired to  pass  10  amperes  through  the  circuit  is  66.7  volts. 
The  complex  expressions  for  these  circuits  reduce  to  the  ordi- 


Ot- 


0,t 


A" 


.10 


A, 


-05_ 

-.16 

Fig.  167.  — Solution  a. 


* 


* The  terms  inductive , capacity,  and  reactive  circuit  are  used  in  this  book 
with  the  following  significations  : an  inductive  circuit  is  one  containing  in- 
ductance, but  not  capacity ; a capacity  or  condenser  circuit  is  one  containing 
capacity,  but  not  inductance ; a reactive  circuit  is  one  containing  either  in- 
ductance or  capacity,  or  both  inductance  and  capacity.  A non-reactive  circuit 
is,  therefore,  one  which  contains  neither  inductance  nor  capacity,  that  is,  one 
which  contains  plain  resistance  only. 


280 


ALTERNATING  CURRENTS 


R,  = 10 
L = .01 


A, 


nary  forms  derived  from  Ohm’s  Law  and  are,  therefore, 
omitted. 

b.  The  circuit  con- 
sists of  a non-reactive 
branch  of  10  ohms 
and  an  inductive 
branch  of  .01  henry 
in  parallel.  The  phase 
diagram  consists  of 
two  lines  at  right  an- 
b=  gles  (one  being  hori- 
zontal), since  the  cur- 
rent in  the  non-reac- 
tive branch  is  in  phase 
A with  the  impressed 
voltage  and  that  in 
the  inductive  branch 
lags  90°  behind.  The  lengths  of  the  lines  are  respectively 

The  vector  poly- 


— (=  .10)  units  and—-  ■(=  .125)  units 
R 2 irfL 

gon  is  as  shown.  The  admittance  of  the  circuit  is  .16  mho  and 
the  impedance  is  6.25  ohms.  The  current  flowing  under  an 
impressed  voltage  of  100  volts  is  therefore  16  amperes,  and  it 
requires  62.5  volts  to  cause  10  amperes  to  flow.  The  angle  6 
is  51°  20'.  The  small  triangle  of  Fig.  168  is  an  impedance 
triangle  drawn  on  a different  scale  from  the  larger  admittance 

- 1 

triangle.  Its  sides  are  derived  from  the  relation  Z = = and 

z = whence  r = and  x=  Y*' 


In  the  complex  expression  we  have  g = l = 

Z - Z“ 


and 


-g  + jb,  therefore 
10 

Fi  = 100  + 0 


F9  = 0 


-JO 

-j 


64  + 0 


Y=  .1  - j .125 


Y = V.l2+  .1252  = .16  mho. 
tan  e = — p e = 51°  20'. 

Z = ^r  = 6. 25  ohms. 


When  100  volts  are  impressed  on  this  branched  circuit, 


SOLUTION  OF  CIRCUITS 


281 


I = EY  = 10  -j  12.5 

and  I = EY  = 16  amperes,  lagging  behind  the  voltage. 

When  10  amperes  flow  through  the  branched  circuit, 

W-iz-  10  - 1 |jL25 

.1-/.125  .0256  . 0256 

and  E = IZ  = 62.5  volts,  leading  the  current. 

The  current  in  each  branch  under  either  set  of  conditions  is 
equal  to  the  admittance  of  the  branch  multiplied  by  the  im- 
pressed voltage  corresponding  to  the  conditions. 

c.  The  circuit  consists  of  a non-reactive  branch  of  10  ohms 
in  parallel  with  an  inductive  branch  having  a resistance  of  10 
ohms  and  an  inductance  of  .01  henry.  The  impedance  and 
the  angle  .of  lag  for  the  inductive  branch  are  found  by  the 
method  given  under  Series  Circuits:  First  Class,  and  the 
admittance  of  the  branch  is  laid  off 
line  making  with  the  horizontal  axis 
an  angle  equal  to  the  angle  of  lag 
taken  backwards.  This  is  line  OA" 
in  the  diagram,  Fig.  169.  The  line 
OA'  represents  the  admittance  of, 
and  the  relative  phase  of  current  in, 
the  non-reactive  branch.  The 
length  and  direction  of  the  line 
0XA 2 in  the  vector  polygon  shows 
the  value  of  the  equivalent  or  joint 
admittance  of  the  circuit  and  the 
angle  by  which  the  phase  of  the 
main  current  lags  behind  the  phase 
of  the  impressed  voltage.  The 
joint  admittance  of  the  circuit  is 
.168  mho  and  the  joint  impedance 
5.95  ohms.  The  current  flowing 
under  an  impressed  voltage  of  100 
volts  is  16.8  amperes,  and  the  voltage  required  to  pass  10 
amperes  through  the  circuit  is  59.5  volts.  The  angle  6 is 
16°  52k  The  triangle  OA! A is  equal  to  the  impedance  tri- 
angle of  the  self-inductive  branch  rotated  on  its  base  until 
its  apex  points  downward.  This  reversed  construction  of  the 


in  the  phase  diagram  on  a 
R,  =io 


R2  = io,  L = .oi 

— ' kfinnrird'- 


A,  -io 

Fig.  169. — Solution  c. 


282 


ALTERNATING  CURRENTS 


impedance  line  (used  also  in  following  problems)  is  made  foi 
convenience  in  obtaining  the  correct  position  of  the  admittance 
line,  since  admittance  is  the  reciprocal  of  impedance  and  taking 
the  reciprocal  of  an  operator  reverses  its  angle.  If  the  position 
of  the  impedance  and  its  components  are  desired,  an  impedance 
triangle  should  be  laid  out  as  in  the  case  of  any  series  circuit. 

The  problem  may  be  worked  out  by  the  use  of  complex 
quantities  as  follows  : 


Y — JLQ A 0 

1 i — i o o J u 

Y = 10 • §_ 

2 100  + 64  100 + 64 

Y=  .161  -j  .049 


Y=  V.l612  + .0492=  .168  mho. 
Z — 5.95  ohms. 

tan  e = e = 16°  52'. 

.161 


When  100  volts  are  impressed  on  the  circuit, 

I—  EY  = 16.1  — j 4.9 

and  /=  EV  = 16.8  amperes,  lagging  behind  the  voltage. 

When  10  amperes  flow  through  the  circuit, 

e=  iz=  ~ = — — + /-MiL 

Y .0282  . 0282 

and  E = 1Z  = 59.5  volts,  ahead  of  the  current. 

The  complex  expressions  for  currents  in  the  branches  when 

100  volts  are  impressed  are 

J1  = .2^  = 100  (.1-/0), 

/j  = 10  amperes,  in  phase 
with  the  voltage, 


Li  = .oi 

L2=.0126 
— ^ (5  Xs*- 


"‘  = tan'1B=0°; 


and 


Y = EY0=  100  (.061 
-i.049), 

72  = 7.81  lagging  behind 
the  voltage, 

= tan-1^?  = 38°  40h 
2 .061 

It  will  be  observed  that  the  arith- 
metical sum  of  the  currents,  10 
amperes  and  7.81  amperes,  in  the 
two  branches  is  greater  than  the  current,  16.8  amperes,  in  the 
main  line. 


A2-l 

Fig.  170.  — Solution  d. 


SOLUTION  OF  CIRCUITS 


283 


d.  The  circuit  consists  of  two  inductive  branches  of  respec- 
tively .01  and  .0125  henry  in  parallel.  The  diagrams  consist 
of  vertical  lines,  as  shown  in  Fig.  170.  The  admittance  is 
.225  mho  and  the  impedance  is  4.44  ohms.  The  current  flow- 
ing when  100  volts  is  impressed  on  the  circuit  is  22.5  amperes, 
and  it  requires  44.4  volts  to  cause  10  amperes  to  flow.  The 
angle  of  lag  is  90°. 

The  complex  expressions  for  admittance  are 
¥1  = -j.  1 Y = .225  mho. 

Y2  = -j.  125  Z=  4.44  ohms. 

f=-jm  0 = tan-^5  = 90". 

0 

The  currents  in  the  branches  when  100  volts  are  impressed  on 
the  circuit  are 

Il  — EYl=.lx  100  = 10  amperes,  lagging  90°, 
and  I2=  EY2  = .125  x 100  = 12.5  amperes,  also  lagging  90°. 
The  main  current  is 


1=  EY  = —j  22.5 

and  1 = 22.5  amperes,  lagging  90°. 

When  10  amperes  are  caused  to  flow  through  the  circuit, 


10 

.225’ 


E = IZ  = 44.4  volts,  90°  ahead  of  the  current. 


two  reactive  branches  of  respec- 


Ri=io,  L,  — .005 
r-OS-5"15Tr"5"'b~'* — 

R =8,  L,  = .0125 


e.  The  circuit  consists  of 
tively  .005  henry 
and  10  ohms,  and 
.0125  henry  and  8 
ohms  in  parallel. 

The  diagrams  are  10 

as  shown  in  Fig. 

171.  The  admit- 
tance of  the  circuit 
is  shown  to  be  .165 
mho,  and  the  im- 
pedance 6.06  ohms. 

The  current  flow- 
ing under  a voltage  of  100  volts  is  therefore  16.5  amperes,  and 
the  voltage  required  to  pass  10  amperes  through  the  circuit  is 


Fig.  171.  — Solution  e. 


284 


ALTERNATING  CURRENTS 


60.6  volts.  The  angle  8 is  35°  16'.  The  problem  may  be 
solved  by  the  use  of  complex  quantities  as  follows : 

y 10  • £_ 

1 100  + 16  100 +16 

Y — § j 10 

2 64  + 100  J 64  + 100 

Y=  .1349  -j.  0954 
Y — ^ . 13492  + .09542  = .165  mho. 

Z = = 6.  06  ohms. 

.165 

tan  6 = = .7072,  6 = 35°  16'. 

.1349 

The  currents  in  the  branches  when  100  volts  are  impressed  on 
the  circuit  are 

Ix  = EYX  — 9.3  amperes,  lagging  behind  the  voltage, 
and  /2  = EY2  = 7.8  amperes,  also  lagging  behind  the  voltage  ; 

dx  = 21°  48' 

and  62  = 51°  20'. 

The  total  current  is 

1=  EY  = 13.5  -j  9.54 

and  I = EY  — 16.5  amperes,  lagging  behind  the  voltage. 

The  five  preceding  examples  cover  all  the  fundamental  com- 
binations of  resistance  and  inductance  in  parallel  circuits.  The 
following  four  in  like  manner  cover  the  combinations  of  resist- 
ance and  capacity.  The  solutions  in  the  two  cases  are  similar, 
but  the  lag  angles  become  negative  in  the  latter  on  account  of 
the  influence  of  the  capacities. 


Circuits  containing  Resistance  and  Capacity 


f.  When  two  or  more  condensers  of  negligible  internal  re- 
sistance are  connected  in  parallel  by  wires  of  negligible  resist- 
ance, they  evidently  act  upon  the  circuit  exactly  as  though 
it  contained  one  condenser  with  a capacity  equal  to  the  com- 
bined capacity  of  those  in  parallel.  The  impedance  of  a con- 


denser is  equal  to 


2 irfC 


, and  its  admittance  to  2 i -fC.  The 


admittance  of  several  condensers  in  parallel  is  therefore 
evidently 

2 Trf{Cx  + C2  + etc.). 


SOLUTION  OF  CIRCUITS 


285 


R - 10 


C = ioo 


A" 


g.  The  circuit  consists  of  a non-reactive  branch  of  10  ohms 
in  parallel  with  a capacity  branch  of  100  microfarads  and  negli 
gible  resistance.  The  dia- 
grams are  as  shown  in 
Fig.  172.  The  admit- 
tance of  the  circuit  is  .128 
mho  and  its  impedance  is 
7.81  ohms.  The  current 
flowing  under  a voltage 
of  100  volts  is  12.8  am- 
peres, and  the  voltage  re- 
quired to  pass  10  amperes 
through  the  circuit  is  78.1 
volts.  The  angle  0 is 
- 38°  40'. 

The  complex  expressions  for  this  case  are  of  the  same  form 
as  for  Solution  b except  that  the  direction  of  the  vertical 
component  is  reversed. 


Fig.  172.  — Solution  g. 


Y^.l-jO 

F„  = o+y.Q8 

Y = .!+/. 08 


Y=  ^.l2  + .082  = .128  mho. 


Z - ——  7.81  ohms. 

6 = — tan-1  = -38°  40'. 


When  100  volts  are  impressed  on  the  circuit, 

I=EY=10+j8 

and  /=  EY  = 12.8  amperes; 

I\  = EYX  = 10  —j  0 

and  Ix  = EYX  — 10  amperes ; 

I2  = EY2=j  8 

and  I2  = EY2  — 8 amperes  ; 

Qx  = 0°,  02  = - 90°. 


When  a current  of  10  amperes  is  caused  to  flow  through  the 
circuit, 

1 . 0.8 

r 


E = — 


.0164  .0164 

E — IZ  =78.1  volts. 


and 


286 


ALTERNATING  CURRENTS 


R = 10 


R =10,  C = ioo 


h.  The  circuit  consists  of  a non-reactive  branch  of  10  ohms 
in  parallel  with  a reactive  branch  of  10  ohms  and  100  micro- 
farads. The  ad- 
mittance of  the  cir- 
cuit is  shown  by 
the  diagrams  of 
Fig.  173  to  be  .117 
mho  and  the  im- 
pedance 6.8  ohms. 
The  current  flow- 
ing under  100 
volts  is  14.7  am- 
peres, and  the  vol- 
tage required  to 
pass  10  amperes 


-19°  20'. 


through  the  circuit 
is  68  volts.  The 
angle  of  lag  is 
The  solution  by  means  of  complex  quantities  is 


Y _ 10  • 0 

1 100  + 0 J 100  + 0 

v 10  ■ 12-5 

2 100  + 156  J 100  + 156 

Y = .1390  +/.  0488 

r = V.l3902+  .04  882  = .147  mho. 

Z = —5—  = 6.8  ohms. 

.147 

tan  ^ = _iM§8  = _.3509,  0 = -19°  20'. 
.1390 


When  100  volts  are  impressed  on  the  circuit,  the  currents  are 
I=EY=  13.9  +i  4.88 


and 


I = EY  = 14.7  amperes; 
Ix  = EY1  = 10  -j  0 
Ix  = EYX  = 10  amperes  ; 


I^EY 


1000  .1250 

256  +J  256 


and 


SOLUTION  OF  CIRCUITS 


287 


and  /2  = E Y2  = 6.25  amperes ; 

6X  = 0°,  02  = — tan"1  = 

When  10  amperes  are  caused  to  flow  through  the  circuit, 


51°  20'. 


E = --  = 


1.39  . .488 

-3 


and  E - IZ  = 68  volts. 

i.  The  circuit  con- 
sists of  two  reactive 
branches  in  parallel,  re- 
spectively, of  10  ohms 
and  100  microfarads, 
and  of  20  ohms  and  250 
microfarads.  The  ad- 
mittance of  the  circuit 
is  shown  by  the  diagram 
in  Fig.  174  to-  be  .105 
mho  and  the  impedance 
9.5  ohms.  The  current 
flowing  under  a voltage 
of  100  volts  is  10.5  am- 
peres, and  the  voltage 
required  to  pass  10  am- 
peres through  the  cir- 
cuit is  95  volts.  The 
angle  6 is  — 35°  10'. 


.0218  " o0218 

R = io,  C=  loo 


R =20, 


C = 260 


R = 7.78 


Fig.  174.  Solution  i. 


The  solution  with  complex  quantities  is 

Y=  A//.08622  + .06052  = .105  mho. 


F=J0  .12,5 

1 256  J 256 

Y=™-+j±- 

2 425  J 425 


Z = —r  = 9.52  ohms. 


Y =.0862+  j.  0605  0 = - tan 


_!  .0605 


= -35°  10'. 


.0862 

When  100  volts  are  impressed  on  the  circuit,  the  currents  are 
I=EY  = 8.62  +j  6.05 
1=  EY  = 10.5  amperes ; 
f 1000  .1250 

I1  = EY1=mt  + 3 


256 


256 


and 


288 


ALTERNATING  CURRENTS 


and 


Ix  = EYX  — 6.25  amperes ; 


i2  = ey2 


2000  .500 

425  + 3 425 


and  I2  = EY2  = 4.85  amperes  ; 

0.  = - tan"1—  = - 51°  20',  02  = - tan^1—  = - 14°  5'. 

1 10  2 20 

When  10  amperes  are  caused  to  flow  through  the  circuit, 

I -862  . .605 

J~Y~  -0111  3 .0111’ 

E = IZ  = 95. 2 volts. 


The  impedance  components  of  the  combined  circuit  may  be 
found  as  in  Problem  b or  from  the  formula 


or 


Z=  --p  (cos  0 +j  sin  0) 
%=  — b (.817  —J  .576), 


Z=  7.78  —j  5.49. 

The  impedance  diagram  is  shown  in  the  triangle  02BC,  Fig. 
174. 

The  following  examples  cover  the  fundamental  combinations  of 
circuits  containing  resistance , capacity , and  self -inductance. 

j.  The  circuit  consists  of  two  reactive  branches  in  parallel, 
respectively  of  5 ohms  and  .005  henry,  and  of  10  ohms  and  100 
microfarads.  The  admittance  of  the  circuit  is  shown  by  the 
diagrams  of  Fig.  175  to  be  .168  mho  and  the  impedance  5.95 
ohms.  The  current  flowing  under  a voltage  of  100  volts  is 
16.8  amperes,  and  the  voltage  required  to  cause  a current  of 
10  amperes  to  flow  is  59.5  volts.  The  angle  6 is  16°  50'. 

The  solution  by  means  of  the  complex  quantities  is  as 
follows : 


Y _ 5 4 

1 25  + 16  J 25  + 16 


Y=  ’s/.1612+  .04872  = .168  mho. 


Y - if  12‘5 

2 100  + 156  '100  + 156 


Z = - ^ = 5.95  ohms. 

.168 

tan  6 = •°487  = .3025,  0 = 16°  50'. 
.1610 


Y = .1610  — j .0487 


SOLUTION  or  CIRCUITS 


289 


When  100  volts  are  impressed  on  the  circuit, 
I = EY  =16.1-/4.87 
and  I = EY  = 16.8  amperes ; 


and 


r 500  .400 

Ix  — EY1  = 15.6  amperes ; 


E = EY„  = 1000 


.1250 


and 


I2  = EY2 


400 


0 = tan-i  = 38°  40',  0O  = 

1 O00  2 


256  ' 256 

6.25  amperes ; 

! 1250 

1000 


tan" 


= - 51°  20'. 


R = 5 , L=  .005 
R = io,  C = loo 


When  10  amperes  are  caused  to  flow  through  the  circuit, 


E=IZ  = 


1.61  . 
.0284 


.487 

.0284’ 


and 


E — IZ  =59.5  volts. 


The  impedance  diagram  for  the  combined  circuit  is  shown  in 
the  triangle  O^BC,  Fig.  175,  and  is  obtained  as  in  the  preced- 
ing cases,  thus : 


290 


ALTERNATING  CURRENTS 


or 


or 


Z=R  + jX, 

Z = y (cos  0 + j sin  0), 


Z = 


.168 

or  Z = Z (cos  6 + j sin  0), 

and  from  either  of  the  last  two, 


(cos  16°  50'  + j sin  16°  50'), 


Z=  5.70  + j 1.72. 


k.  The  circuit  con- 
sists of  two  reactive 
branches  in  parallel, 
respectively,  of  10 
ohms  and  .0156  henry, 
and  of  5 ohms  and  200 
microfarads.  The  ad- 
mittance is  shown  by 
the  diagrams  of  Fig. 
176  to  be  .127  mho, 
and  the  impedance  of 
the  circuit  is  7.87 
ohms.  The  current 
flowing  under  a vol- 
tage of  100  volts  is 
12.7  amperes,  and  78.7 
volts  are  required  to 
pass  10  amperes 
through  the  circuit. 
The  angle  6 is  — 22° 
37'. 


The  solution  using  complex  quantities  is  as  follows : 
10  • 12.5 

1 100  + 156.3  J 100  + 156.3 

r=  5 . 6.25 

2 25  + 39  25  + 39 

Y = .1173  + j .0489 


Y =V.ii732+  .04892=  .127  mho. 

1 


Z = 


.127 


= 7.87  ohms. 


tan  6 = - Q = - 22°  37'. 

.1173 


SOLUTION  OF  CIRCUITS 


291 


and 


and 


When  100  volts  are  impressed  on  the  circuit, 

I—  11.73 +y  4.89 
I = EY  = 12.7  amperes  ; 


r rrtr_  1000  ..1250 

11  256.3  ^ 256.3 

71=  FJ\  = 6.25  amperes; 

7 w 500,  .625 
i = -EF„  = — — 


t2  — ^ ^ 2 

and  I2  = UY2  = 12.5  amperes  ; 

6.25 
5 

= - 51°  20'. 

When  10  amperes  are  caused  to  flow  through  the  circuit. 


6,  = tan  1 ^ = 51°  20',  6 '„  = — tan" 

l io  2 


E=IZ  = 


1.173  . .489 


■J 


and 


.0162  " .0162 
Fj  — IZ=  78.7  volts. 


1.  The  circuit  consists  of  two  reactive  branches  in  parallel, 
respectively,  of  10  ohms  and  .01042  henry,  and  10  ohms  and 


R=10,  L=. 01042 


R =io,  C = iso 


Fig.  177.  — Solution  l. 


150  microfarads.  The  diagrams,  Fig.  177,  show  that  the 
admittance  of  the  circuit  is  .118  mho  and  the  angle  6 is  equal 
to  zero. 


292 


ALTERNATING  CURRENTS 


The  vector  expression  for  admittance  is  as  follows : 


Y 

10 

8.33 

Y— 

1 1 

100  + 69.4  3 

100  + 69.4 

Y2 

10 

8.33 

z — 

100  + 69.4  J 

100  + 69.4 

Y 

= .1181 

3 0 

tan  6 — 

.1181 

0 

.1181 


= 8.47  ohms. 

,6  = 0. 


The  effect  of  the  reactances  in  this  circuit  is  noteworthy. 
It  will  be  observed  that  the  self -inductance  and  capacity  neu- 
tralize each  other’s  effects  so  that  the  angle  of  lag  is  zero,  but 
the  admittance  is  not  much  more  than  one  half  as  great  as 
would  be  the  case  if  the  two  branches  each  contained  10  ohms 
of  resistance  and  no  reactance. 

When  100  volts  are  impressed  on  this  circuit, 


and 


and 


1=  EY  = 11.81  -j  0 
I = EY  = 11.81  amperes; 

j EY  = ^<>(^  — j 833. 
1 1 169.4  J 169.4 

Ix  - EYX  - 7.7  amperes; 

833 

^ — h : v — _i /i 


I0  = EY„=l^r+j 


and 


I,  = EY,= 


169.4  ' 169.4 

7.7  amperes; 


e,= 


tan-1  _833_  _ 39o  48/  Q = tan-i 

1000  2 


R=10,  L=.01 

r^nnnnr^n 

C = ioo 


833 


= -39°  48'. 


1000 

When  10  amperes  are 
caused  to  flow  through 
the  circuit, 

10 


E=IZ  = 


+j  0 


Fig.  178.  — Solution  m. 


.1181 

and 

E = IZ  = 84. 7 volts. 

Under  the  latter  con- 
ditions the  currents  in 
the  two  branches  are, 
respectively,  6.52  am- 
peres and  6.52  amperes. 

m.  The  circuit  con- 
sists of  two  reactive 
branches,  respectively, 


SOLUTION  OF  CIRCUITS 


293 


of  10  olims  and  .01  henry,  and  of  100  microfarads.  The 
diagrams  in  Fig.  178  show  the  joint  admittance  to  be  .0685  mho. 
The  joint  impedance  is  14.6  ohms  the  impedances  of  the 
branches  are,  respectively,  12.8  and  12.5  ohms,  so  that  when 
the  impressed  voltage  is  100  volts,  6.85  amperes  flow  in  the 
main  circuit,  while  7.8  and  8 amperes,  respectively,  flow  in 
the  branches.  The  angle  6 is  — 27°  10'. 

It  will  be  noticed  in  this  case  that  not  only  is  the  arithmetical 
sum  of  the  branch  currents  greater  than  the  main  current,  but 
each  branch  current  is  itself  greater  than  the  main  current. 

The  solution  by  complex  quantities  is  as  follows: 


Y 10  8 

1 100  + 64  3 100  + 64 

Z = 0 +./.08 

Y = .0610  +/.0312 


Y=  ^.OOIO2  + .03122  = .0685  mho. 
Z — 14.6  ohms. 

tand  = -^^=  - .511, 

.0610 

6 = - 27°  10'. 


When  100  volts  are  impressed  on  this  circuit, 


I = EY=  6.1  +j  3.12 

and 

I = EY  = 6.85  amperes; 

T 1000  '800 

1 “ 1 ~ 164  1 164 

and 

Ix  = EY1  = 7.8  amperes; 

T2  = EY2=j  8 

and 

I2  = EY2  = 8 amperes; 

- - tan  1 .8  = 38°  40',  02  = — tan  1 oo  = — 90°. 

When  10  amperes  are  caused  to  flow  through  the  circuit, 

E=IZ  = -•61Q- j ~312  - 

.00469  . 00469 

and  E = IZ  = 146  volts. 

n.  The  circuit  consists  of  two  reactive  branches  in  parallel, 
respectively,  of  .01  henry,  and  of  10  ohms  and  100  microfarads. 
The  diagrams,  Fig.  179,  show  the  admittance  to  be  .0856  mho. 
and  the  impedance  to  be  11.7  ohms.  The  impedances  of  the 
branches  are,  respectively,  8 and  16  ohms,  so  that  when  100 
volts  are  impressed  upon  the  circuit,  8.56  amperes  flow  in  the 


294 


ALTERNATING  CURRENTS 


main  leads,  while  12.5  and  6.25  amperes  flow,  respectively,  in 
the  two  branches.  The  angle  6 is  62°  53'. 


L=  .oi 


Here  one  branch  current  is  much  larger  than  the  main 
current. 

The  solution  by  complex  quantities  is, 


10  12J5 

100  + 156.25  +J  100  + 156.25 

.0391  -j  .0762. 


Y = V.03912  + .07622  = .0856  mho. 
Z = ,.0-„  = 11.7  ohms. 


tan  6 = - 


0856 

0762 

0391’ 


6 = 62°  53'. 


When  100  volts  are  impressed  on  the  circuit, 
1=  EY=  3.91  - j 7.62 
and  1=  EY  — 8.56  amperes; 


SOLUTION  OF  CIRCUITS 


295 


and 


Ix  = EY,=  -j  12.5 
Ix  = EYX  = 12.5  amperes; 

z-^F.-^l+y1250 


tan"1^3  = - 51°  20'. 

1000 


2 2 256.3  ‘ " 256.3 

and  /2  = EY%  = 6.25  amperes; 

0X  - - tan-1  oo  = 90°,  $2  = 

When  10  amperes  are  caused  to  pass  through  the  circuit, 
E = IZ  = 

and 


.391  , . .762 

+J 


....33  ’ " .00733 
E — IZ  - 117  volts. 


The  impedance  diagram  of  the  combined  circuit  02BC  is 
given  in  the  figure  (Fig.  179).  The  sides  of  the  triangle  are 
shown  from  the  expression 

Z = Z (cos  6 + j sin  d) 

Z=  5.34  + j 10.4. 


L = .01 
S = ioo 


TT 

ii 

I O 

• f 


or 

o.  The  circuit  consists  of 
two  reactive  branches  in 
parallel,  respectively,  of  .01  08 

henry  and  of  100  microfar- 
ads. The  diagrams,  Fig.  180, 
show  that  the  admittance 
of  the  circuit  is  .045  mho, 
and  the  admittance  of  the 
branches  are,  respectively, 

.125  and  .08  mho.  When 

the  impressed  voltage  is  100  i =>  /0=9O° 

volts,  the  main  current  is  4.5  A 
amperes,  and  those  in  the 
branches  are  12.5  and  8 amperes,  respectively.  The  angle  6 is  90°. 

The  corresponding  complex  quantities  are: 

- 0.8 

Yx=  - 


.125 


+ - 


±± 


0 + 64 

0 

= 0 + 156 


+.? 


0 + 64 
12.5 
0 + 156 


Fig.  180.  — Solution  o. 


Y = .045  mho 


Y = 


0 -y.045 


Z = 22.2  ohms. 

045 

tan  6 = —7- — = oc  , 0 = 90°. 


296 


ALTERNATING  CURRENTS 


In  this  problem  the  active  (horizontal)  component  of  current 
is  zero,  which  is  a condition  that  can  only  be  approximated  in 
practice.  The  vertical  component  of  current  composes  the  total 
current.  The  branch  currents  are  in  exactly  opposite  phases, 
and  the  main  current  is  equal  to  their  arithmetical  difference. 

When  100  volts  are  impressed  on  this  circuit, 


J = 

EY 

= 

-j  4.5 

and 

I = 

EY 

= 

4.5  amperes; 

EY i 

= 

-y  12.5 

and 

EY, 

= 

12.5  amperes 

I,= 

EY, , 

= 

+ys 

and 

I2  = 

EY, 

= 

8 amperes ; 

6,  = tan-1  oo  = 

90°, 

*2 

= — tan-1  oo 

When  10  amperes  are  caused  to  pass  through  the  circuit, 
E = IZ  = j 222, 

E = IZ  = 222  volts. 


12 


p.  The  circuit  consists  of  two  reactive  branches  in 
parallel,  respectively,  of  .01012  henry  and  150  microfar- 
ads. The  diagrams, 
Fig.  181,  show  that 
the  two  branch  cur- 

S = 160 


L=  .01042 


X 


.12 


A, 


rents  are  in  opposition 
and  of  equal  value, 
and  that  the  admit- 
tance is  zero,  so  that  the  main  current  is  zero. 
A comparison  of  this  problem  with  problem  l 
shows  that  this  condition  can  only  arise  when 
resistance  in  the  branches  is  negligible.  When 
the  impressed  voltage  is  100  volts,  the  branch 
currents  are  each  12  amperes,  and  when  10 
amperes  flow  in  each  branch,  the  voltage  is 
83.3  volts.  No  current  can  be  caused  to  flow 
in  the  main  circuit  leading  to  these  branches.  The  complex 
quantities  representing  the  admittances  are: 

Fx  = 0-y.l2  Y=  0. 

T„  = 0+,M2 


Fig.  181.— 
Solution  p. 


Y = 0 Tj.O 


Z = oo  ohms. 

6 = tan-1.  -J  = indeterminate. 


SOLUTION  OF  CIRCUITS 


297 


If  there  is  any  appreciable  resistance,  however  little,  in  either 
of  these  branches,  Y and  Z have  finite  values  and  6 is  zero. 
In  practical  cases  all  resistance  cannot  be  eliminated,  and  hence 
the  angle  may  be  considered  zero. 

If  circuits  approximating  those  given  in  this  problem  are 
constructed  with  very  small  resistance,  excessively  large  local 
currents  may  flow  when  the  generator  current  is  almost  neg- 
ligible. Such  cases  are  not  unknown  in  practical  operation. 

84.  Conclusions  in  Regard  to  Parallel  Circuits.  — Second 
Class. — - The  sixteen  examples  just  presented  cover  every 
fundamental  arrangement  of  simple  parallel  circuits.  An 
examination  of  the  diagrams  and  the  principles  involved  in 
their  construction  makes  evident  the  following  statements, 
applicable  to  circuits  in  which  the  voltages  and  currents  are 
sinusoidal,  which  are  in  many  respects  analogous  to  those  here- 
tofore given  as  applying  to  series  circuits  : 

1.  When  non-reactive  circuits  are  connected  in  parallel,  the 
total  current  equals  the  arithmetical  sum  of  the  currents  in 
the  branches,  and  the  joint  admittance  of  the  circuit  is  equal 
to  the  arithmetical  sum  of  the  branch  admittances. 

2.  When  inductive  circuits  of  equal  time  constants  are  con- 
nected in  parallel,  the  total  current  equals  the  arithmetical 
sum  of  the  currents  in  the  branches,  and  the  joint  admittance 
of  the  circuit  is  equal  to  the  arithmetical  sum  of  the  branch 
admittances. 

3.  When  inductive  and  non-reactive  circuits  are  connected  in 
parallel  with  each  other,  or  when  inductive  circuits  of  unequal 
time  constants  are  connected  in  parallel,  the  total  current  is 
equal  to  the  vector  sum,  which  is  always  less  than  the  arith- 
metical sum,  of  the  branch  currents,  and  the  individual  branch 
currents  are  each  smaller  than  the  total  current.  The  joint 
admittance  of  the  circuit  is  equal  to  the  vector  sum  of  the 
branch  admittances,  each  of  which  is  less  than  the  joint  total. 

4.  When  condensers  are  connected  in  parallel  by  wires  of  neg- 
ligible resistance , the  total  current  equals  the  arithmetical  sum 
of  the  branch  currents,  and  the  joint  admittance  equals  the  arith- 
metical sum  of  the  branch  admittances. 

5.  When  condensers  are  connected  in  parallel  with  non- 
reactive  resistances , the  total  current  equals  the  vector  sum, 
which  is  always  less  than  the  arithmetical  sum,  of  the  branch 


298 


ALTERNATING  CURRENTS 


currents,  and  the  individual  branch  currents  are  each  smaller 
than  the  total  current.  The  joint  admittance  of  the  circuit 
equals  the  vector  sum  of  the  branch  admittances,  each  of  which 
is  smaller  than  the  joint  total. 

G.  When  condensers  are  connected  in  parallel  with  inductive 
circuits , the  total  current  equals  the  vector  sum,  which  is  always 
less  than  the  arithmetical  sum,  of  the  currents  in  the  branches. 
Since  the  effects  of  capacity  and  of  self-induct.ance,  respectively, 
cause  the  angle  6 to  become  negative  and  positive,  the  individ- 
ual branch  currents  may  be  either  greater  or  less  than  the 
main  or  total  current , depending  upon  the  relation  between  the 
various  capacities  and  inductances  in  the  circuit.  The  joint 
admittance  of  the  circuit  equals  the  vector  sum  of  the  branch 
admittances,  each  of  which  may  be  either  greater  or  less  than 
the  joint  admittance. 

The  third,  fifth,  and  sixth  paragraphs  make  evident  this 
proposition,  which  is  similar  to  that  given  for  series  circuits  : * 
When  in  parallel  circuits  the  currents  in  the  branches  are  all 
either  lagging  or  leading  with  respect  to  the  impressed  voltage, 
the  total  or  main  current  is  always  greater  than  the  current  in 
any  one  of  the  branches.  When  the  currents  in  part  of  the 
branches  lead  the  impressed  voltage  and  in  other  branches  lag 
behind  the  voltage,  some  or  all  of  the  branch  currents  may  be 
greater  than  the  total  or  main  current.  It  is  even  theoretically 
possible  for  the  angles  to  have  such  a relation  that  a large  cur- 
rent may  circulate  in  the  branches,  while  the  main  current  is  zero. 

This  is  obviously  the  result  of  the  opposing  relations  of 
inductive  susceptance  and  capacity  susceptance.  It  may  be 
conceived  that  a local  current  is  caused  to  circulate  between 
the  condenser  and  the  inductance,  under  the  stimulus  of  the 
alternating  impressed  voltage,  acting  to  transfer  energy  back 
and  forth  between  the  capacity  of  the  condenser  and  the 
magnetic  field  of  the  inductance. 

When  the  current  is  not  sinusoidal,  the  deductions  given 
above  do  not  strictly  apply,  but  equivalent  sinusoids  f may 
frequently  be  substituted,  when  the  approximation  of  the  de- 
ductions is  usually  satisfactory.  The  deductions  are  general 
when  applied  to  the  harmonics  of  current  and  voltage  when  the 
latter  are  not  sinusoidal. 


* Art.  82. 


f Art.  90. 


SOLUTION  OU  CIRCUITS 


299 


In  every  case  referred  to  in  these  problems,  it  is  assumed 
that  the  parts  of  the  circuits  have  no  appreciable  mutual  mag- 
netic effect.  If  the  parts  are  mutually  inductive,  the  mutual 
effects  must  be  added  to  those  thus  far  treated,  as  will  be  seen 
in  a later  chapter. 

85.  Solution  of  Parallel  Circuits  by  the  Impedance  Methods.  — 

The  solutions  of  parallel  circuits  may  be  made  by  another 
method  in  which  .voltages 
and  impedances  are  princi- 
pally involved  instead  of 
currents  and  admittance. 

This  method  may  be  readily 
exemplified  by  illustrations. 

Suppose,  for  instance,  it  is 
desired  to  find  the  joint 
impedance  of  the  branched 
circuit  in  example  e (Art. 

83).  It  may  be  assumed 
that  an  impressed  voltage  of  100  volts  acts  on  the  circuit. 
Upon  a line,  OX , representing  this  voltage  (Fig.  182)  is  drawn 
a semicircle.  From  0 draw  the  line  OA  making  a lag  angle  of 

6,  with  OX,  where  tan  6,  = = XXLLa  = .4.  Then  OA  is 

1 1 Rx  Rx 

equal  to  and  XA  is  equal  to  IxXv  since  the  angle  at  A 
is  a right  angle.  The  current  in  this  branch,  when  the  irn- 

OA 

pressed  voltage  is  equal  to  OX,  is  — and  this  may  be  laid 

Ri 

off  from  0 to  B.  The  current  in  the  second  branch  is  given 
by  laying  off  the  direction  of  the  line  OA'  so  that  it  makes 

a lag  angle  of  02  with  OX,  where  tan  02  = ^ — 1.25. 

R2  r2 

The  current  in  the  second  branch  is  equal  to  OA'  divided  by 
R2,  and  when  laid  off  from  0 gives  OB' . The  total  current 
in  the  circuit  is  the  resultant  of  OB  and  OB' , or  OB" . Its 
value  in  amperes  is  16.5.  The  impedance  of  the  circuit  is 


Fig.  182.  — Resultant  Current  in  a Parallel 
Circuit. 


then  — : 
6 = tan-1 


OX  _ 100 
OB"  16.5 


= 6.06  ohms.  The  angle  of  lag  is 
16'.  Figures  183,  184,  185,  186,  187, 


and  188  give  the  solutions  by  the  same  method  for  examples 


-12.5- 


300 


ALTERNATING  CURRENTS 


7=rl®2.  . 5.95 

u 16.8 

Fig.  1S8. 


Fig.  185. 


SOLUTION  OF  CIRCUITS 


301 


5,  c,  d , g , j of  Art.  83.  These  show  fully  the  application 

of  the  method.* 

It  is  seen  that  the  triangles  OXA,  OXA and  OXA"  are 
voltage  diagrams  with  the  apeces  turned  down  for  conven- 
ience ; and  that  they  can  be  converted  into  impedance  dia- 
grams by  dividing  the  sides  by  the  respective  currents  in  the 
branches  and  combined  circuit.  Therefore,  it  is  possible  to  use 
the  complex  quantities  expressing  the  impedances  for  solving 
the  problem.  Thus, 

Zx  = 10  -\-j  4. 

Z2  = 8 +j  10. 

Z1  = VlO2  + 42  = 10.77  ohms. 

Z2  = V82+  102  = 12.81  ohms. 

t E 100  o Q 

1,  — — = — = 9.3  amperes. 

1 Zx  10.77  1 

T E 100  _ q 

J2=^=m8i=7-8amperes- 

61  = tan-1 .4  = 21°  48'. 

62  = tan"1 1.25  = 51°  20'. 

Il  = ij( cos  61  — j sin  dj). 

Jx=  8.62  — j 3.45. 

Likewise,  J2  =.  4.87  — j 6.09. 


The  two  current  vectors  added  give 

1=  13.49  — j 9.54. 


/=  ^13.492  4-  9.542  = 16.5  amperes. 


E 

I 


100 

16.5 


= 6.06  ohms. 


The  angle  of  lag  of  the  main  current  is 

9.54 

6 — tan  1 pgTjT)  = 35°  16'. 

This  is  evidently  more  cumbersome  than  using  the  branch 
admittances  directly,  as  in  the  process  described  in  the  preced- 
ing article. 


* Compare  Bedell  and  Crehore,  Alternating  Currents , p.  292 ; Loppe  et  Bou- 
quet, Courants  Alter natifs  Industriels,  p.  111. 


302 


ALTERNATING  CURRENTS 


R,=  lo,  L,  —.005 
R2=8,  L2=.0125 

BKWTSTinPH 


86.  Series  and  Parallel  Circuits  Combined.  — Third  Class.  — 
Where  series  and  parallel  circuits  are  combined,  the  funda- 
mental  solutions  already  given  apply,  directly,  and  it  simply 
requires  experience  to  acquire  facility  in  the  solutions  relating 
to  the  most  complicated  circuits.  Several  examples  are  given 
below  to  indicate  the  general  procedure.  In  these  examples  it 

is  desired  to  determine  as  before : 

(1)  the  total  impedance  of  the  circuit, 

(2)  the  lag  angle  between  the  total 
current  and  the  impressed  voltage, 

(3)  the  total  current  flowing  under  a 
voltage  of  100  volts,  and  (4)  the  volt- 
age required  to  cause  10 
amperes  to  flow.  The 
frequency  is  taken  nearly 
127J-  cycles  per  second 
so  that  2 7rf  = 800. 

a.  The  circuit  consists 
of  an  inductive  coil 
'x  of  10  ohms  and  .01 
henry  in  series  with 
a branched  circuit  similar  to  e , 
Sect.  83.  We  know  that  the  paral- 
lel part  of  the  circuit  has  an  im- 
pedance of  6.06  ohms,  and  that  the 
lag  angle  is  35°  16b  Therefore  OA 
(Fig.  189)  is  laid  off  in  the  phase 
diagram  6.06  units  in  length,  and  making  the  proper  angle 
with  the  horizontal.  The  line  OA'  is  then  laid  off  horizon- 
tally R3(  = 10)  units  long,  and  OA"  is  then  laid  off  verti- 
cally 2 7t/X3(=:8)  units  long.  In  the  vector  diagram  0 1A1 
is  equal  and  parallel  to  OA , A^2  to  OA'.  and  A^A3  to  OA  . 
The  length  of  the  line  0XA3  gives  the  impedance  of  the  cir- 
cuit, which  is  equal  to  18.8  ohms.  The  current  which  flows 
under  a voltage  of  100  volts  is  5.32  amperes,  and  it  requires 
188  volts  to  cause  10  amperes  to  flow  through  the  circuit. 
The  angle  of  lag,  d,  is  the  angle  A301X.  and  is  equal  to 
37°  34b 

By  means  of  complex  quantities  we  take  first  the  admittance 
of  the  parallel  branches: 


Z =18.8 

Fig.  189.  — Solution  a. 


SOLUTION  OF  CIRCUITS 


303 


Ya  = .086.2  -j  .0345 
Yb  = -0487  -j  .0609 
Tab  = • 1349  -j.  0954 

tan  6 aK  = 0AB  = 35°  16'. 


Z ...  = 


.1349 
1( 

1 


Yab  = .165  mho. 


.165 


= 6.06  ohms. 


The  impedance  Z AB  can  be  divided  into  its  components,  thus 
(note  the  change  of  sign  caused  by  changing  from  admittances 
to  impedances)  : _ 

ZAB  = -J~: 
yab 


.1349  ..0945 

.0272  .0272’ 


or  ZAB  = 6.06(cos  9AB  +j  sin  0AB) 

= 6.06(.816  +j  .577) 

= 4.95  +j  3.50. 

This  impedance  is  to  be  added  to  the  impedance  of  the 
remaining  portion  of  the  circuit,  giving : 

ZJB  = 4.95  +j  3.50 
Zc  = 10  + j 8 

Z = 14.95  +j  11.50 
tan  e = 11^2,  e = 37°  34h 
Z = 18.8  ohms. 

1=  ^^-  =5.32  ampei’es  when  100  volts  are  impressed. 

18.8 

U = 10  x 18.8  = 188  volts  when  10  amperes  flow. 

The  components  of  current  when  100  volts  are  impressed  on 
the  circuit  are 


I=*  = 


100 


£ 14.95  +j  11.50 

= 4.2  -j  3.23; 

that  is,  the  active  current  is  4.2  amperes  and  the  quadrature 
current  is  3.23  amperes. 

When  100  volts  are  impressed  on  the  circuit,  the  current 
flowing  in  the  main  circuit  is  5.32  amperes,  and  the  currents  in 


304 


ALTERNATING  CURRENTS 


the  branches  A and  B are  found  from  the  condition  that  the 
voltage  across  the  branched  part  of  the  circuit  is 

Eab  = ZABI=  6.06  x 5.32  = 32.3  volts. 


and 


or 


The  currents  in  the  branches  are,  therefore, 

4=32.3  4 = 2.78  —j  1.11 
4 = 2.99  amperes; 

4 = 32.3  Yb=  1.57-/1.97, 

4 = 2.52  amperes. 

When  10  amperes  are  caused  to  flow  in  the  main  circuit,  the 
currents  in  the  branches  are  in  the  same  ratio  as  2.99  to  2.52. 

b.  The  circuit  consists  of  an  inductance  of  .01  henry  in  series 
with  a branched  circuit  having  two  parallel  branches  contain- 
ing, respectively,  40  ohms  and  100  microfarads.  The  joint  im- 
pedance of  the  branched  part  of  the  circuit  is  first  found  in 

the  usual  manner. 


40  — R, 


100  = C2 


-72°  40' 


0 -026  A" 


’L3=.01 


This  is  11.9  ohms, 
and  the  lag  is 
- 72°  40'.  In  the 
phase  diagram,  Fig. 
190,  OA!  is  there- 
fore laid  off  equal 
to  11.9  and  making 
angle  of 
40',  and 
laid  off 


a lag 
- 72° 


is 


OA' 

2 7 rfL(  = 8)  units 
in  length  and  mak- 
ing a lag  angle 
of  90°.  Laying  off 
the  vector  diagram 
gives  0XA2  equal  to 
4.9  and  making  a lag  angle  of  — 43°  40'.  The  current  flowing 
under  a voltage  of  100  volts  is  20.4  amperes,  and  the  voltage 
required  to  cause  10  amperes  to  flow  is  49  volts.  When  10 
amperes  flow  in  the  main  circuit,  the  voltage  at  the  terminals 
of  the  branched  circuit  is  119  volts,  and  the  currents  which  flow 
through  the  resistance  and  the  condenser  are,  respectively, 


SOLUTION  OF  CIRCUITS 


305 


3 amperes  and  9.5  amperes.  The  voltage  across  the  inductance 
i3  is  then  80  volts. 

By  complex  quantities  the  joint  admittance  of  the  parallel 

branches  is:  ^ A„r  . „ 

Ya  = .025  +j  0 


Yb  — 0 + j .08 

Yab  = .025  +j  .08 


0ab  = “tan  1 


•uo  _ _ 70°  40'. 
.025 


The  components  of  the  impedance  of  the  branched  circuit 


rab= 

9 _ 

.025 

3.55  ohms, 

Tab 

(.084)2 

XAB= 

b 

.08 

11.38  ohms. 

T AB 

(.084)2 

Therefore  the  impedance  of  the  total  circuit  is: 

ZAB=  3.55  -j  11.38 
Zc=0  +/  8-0 
Z = 3.55  — j 3.38 

Z = V052  + fW  = 4.9  ohms. 

6 = - tan-1  = - 43°  40' . 

The  components  of  the 
branched  circuit  may  be 
obtained  from  the  trigo- 
nometric expression  as 
indicated  in  Solution  a , 
or,  if  more  convenient, 
from  the  logarithmic 
formula. 

bv  If  the  frequency 
in  the  preceding  example 
is  cut  down  to  80,  the 
relations  are  materially 
changed.  The  impe- 
dance of  the  branched  circuit  becomes  17.8  ohms,  and  the  lag 
angle  in  it  is  — 63°  32'.  The  phase  diagram,  therefore,  is  as 


3.55 


306 


ALTERNATING  CURRENTS 


shown  in  Fig.  191.  From  the  vector  diagram  it  is  seen  that 
the  joint  impedance  of  the  whole  circuit  is  13.5  ohms,  and  the 
total  current  is  54°  ahead  of  the  impressed  voltage.  The  total 
current  flowing  when  the  impressed  voltage  is  100  volts  is  7.4 
amperes,  and  it  requires  135  volts  to  cause  10  amperes  to  flow. 
When  10  amperes  are  flowing,  the  voltage  at  the  terminals  of 
the  branched  circuit  is  178  volts,  and  the  currents  which  flow 
through  the  resistance  and  the  condenser  are  4.45  and  8.9 
amperes,  respectively,  while  the  voltage  across  the  inductance 
X3  is  50  volts.  To  maintain  a voltage  of  100  volts  at  the 
terminals  of  the  divided  circuit  requires  a voltage  of  76  volts 
impressed  on  the  total  circuit.  With  this  voltage,  5.6  amperes 
flow  through  the  circuit. 


tan  eAB  = 0AB  = -68°32'. 

.02o 

Y . R = .0561  mho. 


YA  = .025  -/0 
Yri=  0 +/.0503 

Yab  = . 025  +j  .0503 
ZAb  =17.8  (cos  6 AB  -j  sin  0 AB) 
= 17.8  (.4456  -j.  8960) 
Z_AB  = 7.94  -j  15.95 
Zr  =0  +y  5.03 
Z = 7. 94 -j  10.92 


^=^k=17-8ol,ms- 

tan0=  e=  - .54°  O'. 

7.94 

Z = 13.5  ohms. 


/=  ““““7  = 7.4  amperes  when  100  volts  are  imoressed. 
13.5  1 

E = 10  x 13.5  = 135  volts  when  10  amperes  flow. 


When  100  volts  are  maintained  on  the  divided  part  of  the 
circuit, 

IAB  = EYab  = .025  X 100  + j .0503  x 100  = 2.5  + j 5.03 
and  IAB  = ^2.52  + 5.032  = 5.61  amperes. 


The  voltage  measured  between  the  terminals  of  the  part  C 
under  these  conditions  is 


Ec  — Zc  x 5.61  = (0  + j 5.03)  x 5.61, 
Ec  = 28.2  volts, 


and 


0C=  90°. 


The  voltage  impressed  on  the  total  circuit  under  these  con- 

dltlOllS  IS  T,  --  r>  -t  y j j r *4-10 

= o.61  Z = 44.5  —j  61.3, 

E — 5.61  Z — 76  volts. 


and 


SOLUTION  OF  CIRCUITS 


307 


It  will  be  observed  that  the  current,  voltage,  and  angle  of 
lag  in  any  part  of  the  circuit  under  any  conditions  may  be 
computed  by  these  processes  when  the  conditions  are  suffi- 
ciently known. 

c.  The  circuit  consists  of  a combination  as  shown  in  Fig. 
192.  The  resistance  of  the  branches  of  the  circuit  are  R1  = 8 
ohms,  i22=10  ohms,  Rs  = 5 ohms  ; the  inductances  are  Xj  = .0125 
henry,  Z2  = .01  henry,  Lz  = .005  henry  ; and  the  capacities  are 


0^  = 125  microfarads,  (?2  = 100  microfarads,  C3  = 150  micro- 
farads. The  diagrams  show  the  impedance  of  the  branched 
part  of  the  circuit  to  be  4.74  ohms,  and  its  lag  angle  is 
— 10°  12'.  From  the  complete  diagrams  it  is  seen  that  the 
joint  impedance  of  the  whole  circuit  is  10.95  ohms,  and 
the  total  current  is  28°  10'  ahead  of  the  impressed  voltage. 
The  total  current  flowing  when  the  impressed  voltage  is  100 
volts  is  9.12  amperes,  and  it  requires  109.5  volts  to  cause  10 
amperes  to  flow.  When  10  amperes  are  flowing,  the  voltage 
at  the  terminals  of  the  branched  circuit  is  47.4  volts,  and  the 
currents  which  flow  through  the  branches  are  5.9  and  4.3  am- 
peres, respectively.  The  voltage  across  the  first  part  of  the 
circuit  is  then  66  volts.  To  maintain  a voltage  of  100  volts  on 
the  branched  part  of  the  circuit  requires  an  impressed  voltage  of 


308 


ALTERNATING  CURRENTS 


231.1  volts  on  the  whole.  With  this  voltage  21.1  amperes  flow 
through  the  circuit. 

The  calculation  by  complex  quantities  may  be  made  as 
follows : — 

8+/ 10-/ 10  = 8+/0. 

ZB  = 10  +/  8 -/ 12.5  = 10-/ 4.5. 

Hence, 


Ya  = .125-/0 
Yb  =.083+/.  0374 
Yab  = . 208+/.  0374 


= V &2  + y1  = . 211  mho. 
Zab  = -^1  = ohms- 

tan  dAB  = — -,  6ab  = — 10°  12'. 
9 


ZAB  = 4r~=  4.66  — / 0.84 


Y 


Z,  = 


5-/ 4. 33 


Z 

Z 


9.66-/5.17 


= ViZ2  + AT2  = 10.95  ohms. 


9 = tan-1  — = — 28°  10b 

R 


The  main  current  with  100  volts  impressed  between  the  cir- 
cuit terminals  is  ri  C\an  r,PT 

Jh  _ 966  . 517 

“ J~120+‘?  120’ 

/=  ^ = 9.12  amperes. 


While,  with  10  amperes  flowing  in  the  main  circuit, 
E = _ZZ  = 109.5  volts, 


and  the  voltage  at  the  terminals  of  the  branched  circuit  is 
EAB  = 1Z AB  = 47.4  volts. 

The  currents  flowing  in  the  parallel  branches  are  then 
IA  = EAB  Ya  = 5.9  amperes 
and  IB  = EabYb  = 4.3  amperes. 

Also,  the  voltage  across  C is 

Ec—  IZC  = 66  volts. 


SOLUTION  OF  CIRCUITS 


309 


To  obtain  a voltage  of  100  volts  on  A and  B requires  that 


T 100  oi  1 
1 = = 21.1  amperes. 

^ A Ii 


Hence  the  voltage  impressed  on  the  terminals  is 
E=  IZ=  231.1  volts. 


87.  Solution  of  Series-Parallel  Problems  by  the  Impedance  or 
Impressed  Voltage  Method.  — The  second  method  of  solution 
for  parallel  circuits  may  be  applied  to  circuits  like  those  included 
in  the  above  article.  Figure  193  shows  the  solution  for 
example  bx  made  by  that 
method.  In  this  it  is 
assumed  that  100  volts 
are  impressed  upon  the 
branched  part  of  the  cir- 
cuit. Then  lay  off  a 
length  OX  on  the  hori- 
zontal axis  representing  7g 

100  volts  and  mark  Z~  eT6i  = 


Q(J  __  100  _ 9 5 

1_  40""  ’ 


Fig.  193.  - 


■ Solution  of  Problem  &i  by  Voltage 
Diagram. 


which  is  the  current  in  the  first  branch.  The  current  in  the 
second  branch  is  90°  in  advance  of  the  voltage,  and  is  repre- 
sented by  OCv  which  is  vertical  and  2 7rfC2B  ( = 5.03)  units  in 
length.  The  resultant  of  these  currents  is  OC,  which  is  5.61 
units  in  length.  The  impressed  voltage  measured  across  the 
terminals  of  the  entire  circuit  is  the  resultant  of  the  100  volts 
at  the  terminals  of  the  branched  part  of  the  circuit,  and  the 
voltage  required  to  pass  5.61  amperes  through  the  inductance 
Lx  = .01  henry.  The  line  representing  the  latter  voltage  is 
perpendicular  to  the  line  representing  the  current  in  the  circuit. 
Drawing  a semicircle  on  OX , and  from  the  intersection  of  OC 
with  the  semicircle  drawing  a line  to  X gives  the  direction  of 
this  voltage.  The  magnitude  of  the  voltage  is  2 71 -fLxI—  28.2 
volts.  This  must  be  laid  off  from  X to  B , and  the  total  im- 
pressed voltage  is  represented  by  OB.  This  shows  that  when 
100  volts  are  maintained  at  the  terminals  of  the  branched  cir- 
cuit, 76  volts  must  be  impressed  on  the  total  circuit.  The  im- 
pedances of  the  circuit  may  be  calculated  from  the  data  thus 


310 


ALTERNATING  CURRENTS 


found,  as  also  can  the  voltage  required  to  maintain  a certain 
current  through  the  circuit. 

Figure  193a  shows  the  solution  of  example  c by  this  method. 
As  before,  the  voltage  at  the  terminals  of  the  divided  circuit  is 
assumed  to  be  100  volts  for  the  purpose  of  the  solution.  This 
is  laid  down  as  OX , and  a semicircle  is  drawn  upon  the  line  as 
a diameter.  Tan  0A  is  equal  to  zero,  so  that  the  current  in  the 
first  branch  is  laid  off  on  OX  to  B , a distance  of  12.5  units. 


Fig.  194.  — Solution  of  Problem  c by  Voltage  Diagram. 


Tan  6 A = .15,  as  shown  by  calculation,  and  the  line  OA'  is  laid 
off  at  that  angle  from  OX.  From  0 on  this  line,  OB'  is  laid 

off  equal  to  the  current  in  the  second  branch,  or  ^ -OX . The 

resultant  of  the  lines  OB  and  OB'  is  OB’\  which  represents 
the  total  current  in  the  circuit.  The  total  voltage  impressed  on 
the  circuit  is  the  resultant  of  the  voltage  impressed  on  the  divided 
circuit,  the  active  voltage  due  to  resistance  Ry  and  the  reactive 
voltage  due  to  L3  and  (73.  The  active  voltage  required  to  pass 
current  OB"  through  Rz  is  represented  by  0(7,  which  is  equal 
to  IRy  The  reactive  pressure  is  perpendicular  to  this  and 

is  equal  to  2 7 rfL3I  — r- ; it  is  represented  by  the  line  OB. 

- irfCz 

The  voltage  impressed  on  the  parallel  circuit  is  represented  by 
the  line  BE , which  is  equal  and  parallel  to  OX.  The  closing 
line,  OE,  represents  the  impressed  voltage  E on  the  circuit 
when  the  current  is  R and  therefore  the  impedance  of  the  circuit 

isy  = 10.95.  The  angle  of  lag  is  the  angle  COE  = — 28°  10'. 


SOLUTION  OF  CIRCUITS 


311 


87a.  Caution.  — The  foregoing  deductions  of  Arts.  81  to  88 
relate  to  the  impedances  and  admittances  in  circuits  within 
which  the  currents  and  voltages  are  sinusoidal,  and  to  the 
effective  values  of  such  voltages  and  currents.  They  also  relate 
to  the  impedances  and  admittances  encountered  by  single  har- 
monics of  distorted  voltages  and  currents.  That  is,  impedances, 
admittances,  effective  voltages,  and  effective  currents  combine 
vectorially.  But  the  student  must  constantly  remember  that 
instantaneous  values  of  voltages  or  of  currents  at  a junction 
point  must  be  combined  algebraically  since  the  instantaneous 
values  are  simple  algebraic  values  and  the  vector  relations 
heretofore  considered  do  not  extend  to  them.  It  is  frequently 
possible  also,  for  approximate  solutions,  to  substitute  equivalent 
sinusoids  * for  the  irregular  curves,  though  on  account  of  the 
extreme  activity  which  is  sometimes  exhibited  by  the  higher 
harmonics  f — especially  where  a condition  of  resonance  is  ap- 
proached — each  problem  should  be  carefully  studied  before 
such  a substitution  is  made. 

* Art.  90.  t Art.  69. 


CHAPTER  VII 


POWER,  POWER  FACTOR 


88.  The  Power  expended  in  a Circuit  on  which  a Sinusoidal 
Voltage  is  Impressed. — a.  Non-reactive  Circuit.  In  a circuit 
without  inductance  or  capacity,  the  current  agrees  in  phase 
with  the  voltage  which  sets  it  up.  The  rate  of  expenditure  of 
energy  in  the  circuit  at  any  instant  is  equal  to  the  product 
of  the  corresponding  instantaneous  current  and  voltage.  The 
average  rate  of  expenditure  of  energy,  or  the  average  value  of 
the  power  expended  in  the  circuit,  during  a complete  period  is 
equal  to  the  average  of  all  the  instantaneous  products.  Or, 


ie  At , but  with  the  voltage  and  current  sinusoidal 


e 


em  sin  a and  i = 


R' 


where  T is  the  time,  of  a period  and  em  is  the  maximum  voltage ; 
hence,  P = J ^ ie  At  = sin2  ada  = AA, 

but  E = -*0-  and  I = ^ ==  ■ 

V2  E V2  R 

Hence,  P = ^ = IE, 

I and  E being  the  effective  values  of  the  current  and  voltage. 

b.  Reactive  Circuit.  If  the  circuit  under  consideration  is 
reactive,  the  current  is  caused  to  lag  behind  or  lead  the  voltage 
by  the  angle  6.  The  rate  of  expenditure  of  energy  in  the  cir- 
cuit at  any  instant  is  evidently  still  equal  to  the  product  of  the 
corresponding  instantaneous  values  of  the  current  and  voltage. 

312 


POWER,  POWER  FACTOR 


313 


The  expression  for  the  average  power  expended  in  the  circuit 
is,  therefore,  as  before, 

P — 2 ie  A t. 

o 

In  this  case,  however,  with  the  voltage  and  current  sinusoidal 
e = em  sin  a,  and  i sin  (a  — 6 ).  * 

Hence,  P =•%=  f sin  a sin  (a  — 0)da  — -m  c<^  ^ f sin2  ada 
ttZJo  nZ  Jo 

e j sin  Of71.  , ej  cos  0 

- — — — I sin  a cos  ada  = — — — — , 

7 tZ  2 Z 


and  since  em  = _EfV2,  and  — = IV 2,  there  follows 

jP  = IE  cos 


Also 


cos  6 = 


P 

IE 


Since  sinusoidal  current  and  voltage  curves  have  equal  posi- 
tive and  negative  loops,  the  expression  thus  derived  for  the 
power  expended  in  a circuit  during  one  half  period  applies  to 
every  half  period,  and  therefore  to  continuous  operation.  In 
the  ordinary  measurements  of  current  and  voltage  the  effective 
values  of  the  quantities  are  determined.  Consequently,  the 
product  of  amperes  and  volts , thus  determined,  does  not  represent 
the  powqr  expended  in  a reactive  circuit , but  the  product  must  be 
multiplied  by  the  cosine  of  the  angle  of  lag.  On  the  other  hand, 
a Wattmeter,  that  is,  an  electrodynamometer  with  one  coil  of 
low  resistance  connected  in  series  with  the  circuit  and  another 
coil  of  high  resistance  connected  in  shunt  with  the  circuit,  aver- 
ages the  instantaneous  products,  and  therefore  gives  readings 
that  are  directly  proportional  to  the  power  absorbed. 

The  preceding  expressions  show  that  the  power  expended  in 
a reactive  circuit  is  equal  to  the  voltage  E,  multiplied  by  a 
component  of  the  current  which  is  in  phase  with  the  voltage 
and  is  equal  to  I cos  0.  This  component  may  be  called  the 
Active  or  Energy  current.  The  remaining  rectangular  compo- 
nent of  current,  I sin  6 , is  in  quadrature  (that  is,  at  90°  differ- 
ence of  phase)  with  the  voltage  and  does  not  contribute  to  the 


* Arts.  58  c and  62. 


314 


ALTERNATING  CURRENTS 


power  expenditure  when  taken  through  a complete  period. 
This  quadrature  component  which  does  no  work  during  a full 
period  may  be  called  the  Wattless  or  Quadrature  current.  For 
illustration,  suppose  in  Fig.  195  that  OE  is  the  vector  voltage 

impressed  on  a cir- 
cuit, 01  is  the  cur- 
rent, and  9 the  angle 
of  lag.  Resolving 
01  into  its  compo- 
nents in  consonance 
and  in  quadrature 
with  the  voltage, 
gives  0Ia  = 01  cos  9 
and  0IX  = 01  sin  9. 
Multiplying  the 
former  by  E gives 
the  power  expended, 
El  cos  9 , and  the 
component  OIx  is  therefore  inactive  when  averaged  over  a 
whole  period. 

Since  instantaneous  power  is  equal  to  ei,  it  is  obvious  that 
quadrature  current  must  represent  work  done  during  some  part 
of  the  period,  and  that  the  wattless  character  must  therefore 
be  due  to  alternate  absorption  and  return  of  power  by  the 
circuit.  Taking  the  summation  of  eiAt  for  successive  quarters 
of  a period  proves  this.  Thus, 


Fig.  195.  — Vector  Diagram  of  Current  Components 
and  Voltage. 


2 P 2 C'l 
7 tZ  ‘/0 

and 


r sin  a sin  («  y 90 °)da  = — f sin  a sin  («  T 90 °)da, 
•sn  7 rZ 

2 

f sin  a sin  (a  T 90°)d«  = 0. 

7T  Z Oo 


This  shows  that  the  quadrature  current  is  inactive  only  when 
considered  for  any  consecutive  length  of  time  equal  to  a full 
half  period.  It  in  fact  is  a vehicle  of  energy  which  swashes 
back  and  forth  in  the  circuit,  in  one  quarter  period  chargiug 
up  the  magnetic  or  electrostatic  fields  and  in  the  next  quarter 
period  discharging  the  fields  and  returning  the  energy  to  the 
source,  the  next  quarter  period  charging  the  fields  in  the 
opposite  direction,  followed  again  by  the  return  of  the  energy 


POWER,  POWER  FACTOR 


315 


to  the  source  in  the  succeeding  quarter  period,  thus  balancing 
off  the  ebb  and  flow  of  energy  in  each  half  period  and  calling 
for  no  continuous  expenditure  of  power. 

If  0 were  90°,  the  total  current  would  be  in  quadrature  with 
the  voltage  and  therefore  inactive.  This  would  be  possible  only 
in  a circuit  having  no  electrical  resistance  or  other  grounds 
for  conversion  of  electrical  energy  into  heat,  light,  or  mechani- 
cal power ; otherwise  some  power  would  necessarily  be  ex- 
pended in  heating  the  conductors,  etc.,  and  the  current  would 
have  to  include  an  energy  component.  It  is  possible,  how- 
ever, to  make  the  ratio  of  inductive  reactance  2 nfL  so  great 
in  comparison  with  the  resistance  R,  by  using  special  inductive 
coils,  that  the  angle  0 is  very  nearly  90°.  It  is  also  possible  to 
make  the  capacity  of  a circuit  so  great  in  comparison  with  the 
resistance  that  the  angle  6 is  nearly  — 90° ; and  thereby  these 
deductions  in  regard  to  wattless  current  may  be  tested  in  the 
laboratory. 

Prob.  1.  The  current  flowing  in  a circuit  is  65  amperes 
and  the  voltage  is  200  volts ; the  angle  of  lag  between  the  cur- 
rent and  voltage  is  60°.  What  is  the  power  expended  in  the 
circuit?  Would  a change  of  frequency  change  the  power,  the 
voltage,  current  and  angle  of  lag  remaining  the  same? 

Prob.  2.  Three  reactive  coils  respectively  comprising  resis- 
tance and  self- inductance  as  follows:  4 ohms  and  .01  henry, 
8 ohms  and  .02  henry,  and  10  ohms  and  .05  henry,  are  con- 
nected in  series.  Fifty  amperes  flow  in  the  circuit.  If  the 
frequency  is  60  periods  per  second,  how  much  power  is  ex- 
pended in  the  total  circuit,  and  how  much  in  each  part  ? 

Prob.  3.  An  overhead  transmission  line  has  a resistance  of 
2 ohms  and  a self-inductance  of  .0002  of  a henry.  This  circuit 
supplies  fully  loaded  transformers  which  in  effect  give  a non- 
inductive  load  which  has  an  equivalent  resistance  of  10  ohms. 
If  the  voltage  at  the  receiving  end  where  the  transformers  are 
placed  is  1000  volts  when  the  frequency  is  60  periods  per  second, 
what  is  the  voltage  at  the  generator,  how  much  power  is  lost  in 
the  line,  and  how  much  is  utilized  by  the  transformers  ? 

Prob.  4.  The  three  reactive  coils  of  Prob.  2 are  placed  in 
parallel,  and  100  volts  with  a frequency  of  60  periods  per 
second  are  impressed  between  their  terminals.  What  is  the 
total  power  expended  in  the  circuit? 


316 


ALTERNATING  CURRENTS 


89.  Power  in  a Circuit.  An  Exercise.  — Blakesley  has  given 
a graphical  proof  of  the  formula  P = IE  cos  9 when  I and  E 
are  sinusoidal,  which  is  presented  here  for  the  purpose  of  an 
exercise.  Applying  the  graphical  representation  of  alternating 
voltages  and  currents  by  means  of  rotating  lines,  let  AB  and 
AO  (Fig.  196)  represent  respectively  the  maximum  value  of 

the  impressed  voltage  in  a 
circuit  and  the  maximum 
value  of  the  resulting  cur- 
rent. The  angle  BAC  is 
the  angle  of  lag.  If  the 
lines  rotate  counter-clockwise 
about  the  point  A,  the  in- 
stantaneous projections  of 
the  lines  AB  and  AC  upon 
the  axis  of  Y represent  the 
instantaneous  values  of  the 
voltage  and  current,  when  « is  measured  from  the  X axis.  It 
is  therefore  desired  to  determine  the  average  value  of  the  prod- 
ucts of  these  projections  for  the  purpose  of  obtaining  the  average 
power  for  a period.  Draw  AB'  and  AC  respectively  perpen- 
dicular and  equal  to  AB  and  AC.  These  lines  represent  the 
positions  of  AB  and  AC  after  revolving  through  90°.  In  the 
figure,  the  angle  BAX  represents  a,  and  CAX  represents  a—  9. 
Also,  the  angle  B' AY  = BAX , and  CAY  — CAX.  It  is  then 
seen  from  the  figure  that 

AE  x AD  = AC  sin  CAX  x AB  sin  BAX, 
or  ie  = im  sin  (a  — d)  x em  sin  « ; 

and  in  the  same  way 

AE'  x AD'  = A C'  cos  CA  Y x A B'  cos  B'A  Y, 
i'e'  = im  cos  (a  — 6)  X em  cos  a. 

The  mean  of  these  expressions  is 

IFrhU-  — hAjn  a sjn  _ Qy  _|_  cog  a cos  Qa  _ J 

_ l_m^m  cog  (a  — 0)]  = lCm.  cos  9 = IE  cos  9. 


Fig.  196.  — Voltage  and  Current  Vectors. 


or 


POWER,  POWER  F ACTOR 


317 


This  is  the  expression  for  tire  mean  of  the  products  of  e and  * 
for  two  values  of  « which  are  90°  apart.  This  mean  value  is 
independent  of  the  positions  of  the  lines  in  the  figure,  and 
is  therefore  the  mean  for  all  positions. 

90.  Power  expended  in  a Circuit  when  the  Voltage  and  Cur 
rent  are  Any  Single-valued  Periodic  Functions.  — If  the  current 
in  a circuit  is  not  sinusoidal  and  is  derived  from  a periodic 
voltage  of  irregular  form,  the  instantaneous  values  of  current 
and  voltage  are  * 


* = V sin  («  + /3j)  + im%  sin  (2  « + /32)  + im%  sin  (3  « + £„)  — 


and  e = em^  sin  (a  + 7l)  + e„h  sin  (2  « + 72)  + em%  sin  (3  « + 73)  • • • 
+ emn  sin  (na  + 7,,) . 


Also,  ie  is  the  instantaneous  value  of  power,  and  - eidt  is  the 
average  power ; therefore, 


+ J0  imfm,  sin  (3  a + #3)  sin  (3  a + 73)  da  — 

+ f0  imnemn  sin  (na  + Bn)  sin  (%  « + 7n)  da  . 

All  the  other  terms  obtained  by  multiplying  the  values  of  i 
and  e together  are  omitted  as  they  become  zero  when  integrated 
between  0 and  2 7r.f 

Expanding  and  integrating  the  above  expression  gives, 


+ *m,  sin  (na  + £„), 


sin  (a  + £1)  sin  (“  + 7i)  da 


p = l imemi  cos  (/3j  - 71)  + \ SS  c°s  (/32  - 72) 

+ I hnf  rn,  COS  (B,  ~ 7a)  • ■ • • + i COS  (/3„  - Jn). 


* Art.  G7. 


t Art.  13. 


318 


ALTERNATING  CURRENTS 


Since  each  /3  represents  the  angular  displacement  of  a current 
harmonic  from  the  zero  of  time  and  each  7 represents  the  angu- 
lar displacement  of  a voltage  harmonic  from  the  zero  of  time,  it 
is  apparent  that  each  (/3  — 7)  represents  the  angular  difference 
of  phase  between  the  voltage  and  current  for  the  particular 
harmonic ; that  is,  it  corresponds  with  the  angle  of  lag  9 associated 
with  the  frequency  of  the  particular  harmonic.  Therefore, 
since  1 imem  = IE , 

P=I1E1  cos  9X  + I2E2  cos  9.,  4-  IZEZ  cos  03  + •••  InEn  cos  6n. 

Each  term  to  the  right  in  this  formula  is  the  power  expended 
in  the  circuit  by  the  particular  harmonic  and  is  independent  of 
the  other  harmonics.  The  total  power  expended  in  the  circuit 
is  therefore  the  sum  of  the  power  independently  expended  b}r 
the  several  harmonics.  This  theorem  also  follows  from  the 
fact  that  the  product  of  rotating  vectors  differing  in  frequency 
is  equal  to  zero,  and  hence  ImEn  = 0,  where  m and  n are  unequal 
integers.  It  will  be  observed  that  the  power  expended  in  a 
circuit  when  the  current  and  voltage  are  sinusoidal  may  be  de- 
rived from  the  foregoing  formula,  since,  in  that  case,  all  of  the 
higher  harmonics  are  zero. 

When  9 = 90°  for  any  harmonic,  that  harmonic  is  inactive. 
Also  if  a particular  harmonic  of  current  is  present  but  the 
corresponding  harmonic  of  voltage  is  zero,  the  current  harmonic 
is  obviously  inactive ; and  likewise,  if  a voltage  harmonic  is  not 
accompanied  by  a current  harmonic  of  the  same  frequency,  the 
voltage  harmonic  does  not  contribute  to  average  power,  though 
it  may  contribute  to  the  energy  that  swashes  back  and  forth  in 
the  circuit  and  balances  out  for  each  half  period. 

The  effective  current  squared  in  a circuit  is 


"2*-  = I\  + 1%  + I3  + "•  In-> 


and  the  power  expended  in  heating  a conductor  of  resistance  R 
when  current  I flows  is  represented  by 

ER  = I?  R + J22  R + I*  R + - I*  R. 


* Art.  67. 


POWER,  POWER  FACTOR 


319 


The  active  voltage  in  the  circuit  may  be  written  for  convenience 
E cos  0,  which,  substituted  for  its  equivalent  IR  in  the  last 
formula,  gives  as  before, 

P = IE  cos  0=1^  cos  0l  + I2E2  cos  02  + I3 E3  cos  03  + etc. 
With  irregular  waves  the  phase  difference  0 cannot  be 

p 

measured  directly  from  the  plotted  curves,  but  0 = cos-1  — . 

IE 

It  is  sometimes  convenient  to  use  the  formula  P = IE  cos  0 in 
place  of  the  more  complicated  formula  P = IlE1  cos  0X  + etc., 
and  in  that  case  I and  E may  be  thought  of  as  like  sinusoidal 
functions  differing  in  phase  by  the  angle  0.  When  so  con- 
sidered I and  E are  called  Equivalent  sinusoids.  Under  some 
circumstances  equivalent  sinusoids  can  be  substituted  for  com- 
pound curves  without  error,  but  as  harmonics  of  different 
frequencies  differ  in  their  effects  on  circuits  containing  self- 
inductance and  capacity,  the  substitution  must  be  made  with 
discretion. 

91.  Alternating  and  Pulsating  Functions.  — It  has  heretofore 
been  pointed  out  that  commercial  alternating  currents  and 
voltages  ordinarily  comprise  harmonics  of  only  odd  numbers  of 
times  the  fundamental  frequency,  namely,  1,  3,  5,  7,  etc.,  in 
which  case  the  successive  half  cycles  are  like  in  form;  but  there 
are  nevertheless  many  instances  in  which  harmonics  appear  of 
even  numbers  of  times  the  fundamental  frequency.  The  fore- 
going demonstration  has  therefore  been  made  general  in  respect 
to  the  frequencies  of  the  harmonics,  but  the  following  dis- 
cussion will  be  confined  mostly  to  the  formulas  with  the 
even-numbered  harmonics  omitted.  These  may  easily  be  re- 
introduced, however,  for  use  in  connection  with  any  particular 
problem  requiring  their  presence.  It  is  also  to  be  noted  that 
the  introduction  of  a constant  term  into  the  general  formulas 
causes  a larger  average  current  to  flow  in  one  direction  than  the 
other  or  converts  the  alternating  functions  into  pulsating  func- 
tions. Then, 

I2  = 702  + I*  + /22  + I2  + - In\ 

E2  = E2  + E*  + E22  + E2  + - In2 ; 

the  power  expended  in  heating  a conductor  of  resistance  R,  by 
I2R  = I2R  + Iy-R  + I2R  + I2R  + • InR \ 


320 


ALTERNATING  CURRENTS 


and,  P = IE  cos  8 = I^E^  + I\P\  cos  8l  + J2^72  cos  02 

+ I3Es  cos  03  + •••  InEn  cos  0n ; 

in  which  J0  and  E0  respectively  represent  the  average  values  of 
the  current  and  voltage  for  a whole  period. 


92.  Power  Loops. — - Power  loops  or  curves  may  be  plotted  as 
in  Figs.  197  to  201,  in  which  the  ordinates  of  the  dotted  curves 

represent  the  prod- 
ucts of  the  corre- 
sponding ordinates  of 
the  current  and  vol- 
tage curves  (assumed 
sinusoidal)  plotted  in 
full  lines.  Figure  197 
is  for  a non-inductive 
circuit  in  which  the 
voltage  and  current 
reverse  their  direc- 
tions at  the  same 
time,  and  the  power 
ordinates  are  there- 

Fig.  197. -Power  Loops  in  Non-reactive  Circuit.  fore  always  positive, 

but  their  numerical  values  vary  in  each  half  period  from  0 to 
imem  and  back  to  0,  so  that  the  power  absorbed  by  the  cir- 
cuit varies  continu-  + + 

ally  during  each  half 
period.  In  this  case 
6 — 0,  cos  0 = 1 , and 
the  average  power  is 
P = IE.  Figure  198 
shows  the  power 
loops  for  a reactive 
circuit  in  which  the 
angle  of  lag  is  45°. 

This  may  be  taken  to 
represent  equally  the 
condition  when  the 
current  leads  or  lags.  It  will  be  seen  in  this  case  that  during 
a portion  of  each  half  period  the  current  and  voltage  are  in 


POWER,  POWER  FACTOR 


321 


Fig.  199.  — Power  Loops  in  a Circuit  having  a 90°  Lag. 


opposite  directions,  and  some  of  the  ordinates  of  the  power 
loops  are  therefore  negative.  This  must  always  be  the  case 
when  the  current  and  voltage  do  not  coincide  in  phase.  During 
the  portion  of  the  half  period  in  which  the  ordinates  of  the 
power  loops  are 
positive  the  circuit 
absorbs  energy,  but 
during  the  portion 
in  which  the  ordi- 
nates are  negative 
the  circuit  gives  out 
energy  which  was 
stored  as  magnetic 
field  or  condenser 
charge  and  returns 
it  to  the  source. 

The  total  work  given  to  the  circuit  during  the  half  period  is 
equal  to  the  difference  of  that  represented  by  the  area  of  the 
positive  loop  and  that  represented  by  the  area  of  the  negative 
loop,  and  the  average  power  absorbed  by  the  circuit  is  equal 

to  this  difference 
divided  by  the 
length  of  the  half 
period.  When 

0 = 45°, 

P=IE  cos  45° 
= .707  IE. 

When  the  ordi- 
nates of  the  loops 
represent  prod- 
ucts of  instanta- 
neous volts  and 
amperes,  the 
areas  represent  joules,  and  the  areas  divided  by  their  respective 
lengths  measured  in  seconds  give  average  power  in  watts. 

Figure  199  shows  the  power  loops  for  a circuit  in  which  the 
current  and  voltage  differ  in  phase  by  90°.  Tn  this  case  the 
negative  loops  are  equal  to  the  positive  ones,  or  the  circuit  and 
source  alternately  give  and  take  equal  amounts  of  energy,  so 


Fig. 


200.  — Power  Loops  showing  Negative  Work. 
Current  lags  135°. 


322 


ALTERNATING  CURRENTS 


that,  taking  each  half  period  as  a whole,  no  energy  is  absorbed 
by  the  circuit.  In  this  case  cos  6 = 0,  and  therefore  P = 0. 
Figure  200  shows  the  power  loops  when  the  current  lags  135°. 

Under  these  circum- 
stances the  power  is 
negative,  i.e.  the  cur- 
rent is  working  against 
the  induced  voltage,  as 
in  a motor. 

In  case  the  curves  of 
voltage  and  current 
are  not  sinusoidal,  the 
power  loops  are  not 
symmetrical.  Such 
loops  are  shown  by 
the  broken  lines  in 
Fig.  201. 

Power  loops  are 
periodic  curves  of 
double  the  frequency  of  the  voltage  and  current  curves,  since 
they  are  obtained  by  multiplying  current  and  voltage  together, 
and  the  argument  of  the  product  is  equal  to  «+(a  — d)  = 2 u — 6. 
That  is,  a cycle  of  the  power  loops  is  completed  for  each  half 
period  of  current  and  voltage.  The  power  loops  may  be 
expressed  by  means  of  Fourier’s  series  in  the  same  manner  as 
any  other  single-valued  periodic  curve;  thus, 


Fig.  201.  • — Power  Loops  derived  from  Irregular 
Voltage  and  Current  Curves. 


p = P + p'm  sin(2 a + 8a)  + p"m  sin (4  a + S2)  + etc.* 

Where  p is  the  instantaneous  power  corresponding  to  the 
angle  a,  P is  the  net  average  power  taken  over  a period,  and 
p' mi  p" mi  etc.,  are  the  amplitudes  of  the  harmonics. 

Prob.  1.  A circuit  has  a resistance  of  8 ohms  and  a self- 
inductance of  .01  of  a henry  in  series.  The  voltage  impressed 
is  100  volts,  at  a frequency  of  120  cycles  per  second.  Con- 
struct the  power  loops,  assuming  the  voltage  to  he  sinusoidal. 

Prob.  2.  A circuit  has  a resistance  of  20  ohms,  a capacity  of 
400  microfarads,  and  a self-inductance  of  .01  henry  in  series. 
Draw  the  power  loops  produced  when  a sinusoidal  voltage  of  200 
volts  at  a frequency  of  40  cycles  per  second  is  impressed  thereon. 


* Art.  12. 


POWER,  POWER  FACTOR 


323 


Prob.  3.  A circuit  of  100  microfarads  capacity  and  5 ohms 
resistance,  lias  a sinusoidal  voltage  of  50  volts  at  a frequency 
of  60  periods  per  second  induced  in  it.  A sinusoidal  current 
of  50  amperes  is  caused  by  an  external  impressed  voltage  to 
flow  through  the  circuit  in  exact  opposition  to  the  induced 
voltage.  What  is  the  magnitude  and  relative  phase  of  the 
impressed  voltage  ? Is  the  power  positive  or  negative  ? Con- 
struct the  power  loops. 

93.  Trigonometrical  Proof  that  Power  Curves  produced  by  Sin- 
usoidal Voltages  and  Currents  are  Double  Frequency  Sinusoids 
with  their  Axes  Displaced.  — Sinusoidal  current  and  voltage  of 
equal  frequency  in  a circuit  having  any  fixed  phase  relation 
produce  instantaneous  power  that  may  in  general  be  expressed 

P = «»*»  sin  « sin  (a  - 0), 

where  0 is  the  angle  of  lag.  This  expanded  gives 
P = erJm  [sin2  a cos  9 — sin  a cos  a sin  0] . 

The  trigonometrical  part  of  the  expression  on  the  right  may  be 
written  (i  _ 1 cos  2 «)  cos  9 — (1  sin  2 a)  sin  9 

= 1 cos  9 — I (cos  2 a cos  9 + sin  2 « sin  01 
= | cos  9 — \ cos  (2  a — 9). 

Hence,  the  instantaneous  power  is 

P = erJm  (I  COS  0 - | COS  (2  rt  - 0)), 
p = El  cos  0 — El  cos  (2  « — 0). 

The  second  term  in  the  right-hand  member  of  this  equation  is 
a sinusoid  having  its  maximum  ordinate  equal  to  AT,  the  appar- 
ent power.  It  is  displaced  with  reference  to  the  voltage  curve. 
Since  the  instantaneous  values  depend  on  2 a,  it  is  obvious 
that  the  curve  goes  through  two  cycles  in  the  period  of  time 
required  for  the  current  and  voltage  in  the  circuit  to  go 
through  one  cycle,  and  the  power  therefore  has  double  the 
frequency  of  the  voltage  and  current  curves. 

The  first  term  in  the  right-hand  member  of  the  last  equation 
depends  upon  the  cosine  of  the  lag  angle,  which  is  assumed  to 
be  constant.  This  constant  term  is  added  algebraically  to  the 


324 


ALTERNATING  CURRENTS 


ordinates  of  the  cos(2  a — O')  curve  of  the  second  term,  and  there- 
fore displaces  the  horizontal  axis  of  symmetry  of  the  power  loops 
above  or  below  the  axis  of  abscissas  of  the  current  and  voltage 
curves.  Figure  200  illustrates  the  arrangement  where  the  vol- 
tage curve  E has  an  effective  value  of  100  volts,  and  the  current 
curve  / has  an  effective  value  of  30  amperes  and  lags  135°.  The 
axis  of  the  power  curve  P is  YYr,  and  is  below  the  axis  X in  the 
scale  of  watts  a distance 

///cos  0 = 100  x 30  cos  135°  = - 2121, 
which  is  the  average  power  expended  in  the  circuit. 

When  the  angle  TO  < 90°,  the  value  of  El  cos  0 is  positive, 
and  the  average  power  is  positive,  i.e.  is  absorbed  by  the 
circuit,  and  when  t 0>9O°,  as  in  the  example  above,  the  power 
is  negative,  i.e.  is  delivered  by  the  circuit.  When  6 = 0,  the 
equation  becomes  p = EI(1  — cos  2 a),  in  which  case  the  dis- 
placement of  the  axis  is  maximum  and  the  power  loops  just 
touch  the  axis  of  abscissas  as  illustrated  in  Fig.  197.  When 
6 =90°,  p = — El  sin  2 a and  the  axis  of  symmetry  of  the  power 
loops  corresponds  with  the  axis  of  abscissas,  as  shown  in 
Fig.  199. 

An  examination  of  the  power  curves  shows  that  an  alternator 
which  is  connected  into  a highly  reactive  circuit,  or  one  in 
which  the  phases  of  current  and  voltage  are  far  apart,  may  be 
under  a serious  strain  even  though  it  may  be  furnishing  very 
little  power  to  the  circuit.  This  follows  from  the  fact  that  for 
one  quarter  cycle  the  machine  may  be  delivering  large  power  to 
the  circuit  and  during  the  next  quarter  cycle,  when  the  power 
is  negative,  the  circuit  may  be  returning  nearly  as  much  power 
to  the  machine,  tending  to  drive  it  as  a motor.  This  alternate 
generator  and  motor  action  due  to  the  positive  and  negative 
loops  is  of  special  importance  when  there  are  alternators  in  the 
circuit  running  in  parallel  or  as  synchronous  motors.* 

Prob.  1.  Using  the  formula  given  on  page  323,  draw  the 
power  loops  for  a sine  voltage  of  200  volts  and  a sine  current  of 
50  amperes  when  the  current  lags  90°.  Obtain  the  power  loops 
for  the  same  current  and  voltage  when  the  lag  is  45°,  90°.  135°, 
and  180°,  0°,  — 45°,  — 90°,  and  — 135°,  and  compare  the  sets  of 
loops  with  each  other. 


Chap.  XII. 


POWER,  POWER  FACTOR 


Prob  2.  Draw  the  power  loops  in  which  the  current  and 
voltage  are  expressed  by  the  formulas  i = 100  sin  («  + 30°) 
-f  10  sin  (3  a + 60°),  and  e = 200  sin  a + 15  sin  3 a + 5 sin  5 a. 


94.  Apparent  Power  ; True  Power ; Power  Factor.  — The 

product  of  the  effective  values  of  the  current  and  voltage , IE,  in  a 
reactive  circuit  is  called  the  Apparent  Power  or  Volt- amperes  in 
the  circuit.  The  term  Kilo-volt-amperes  is  also  frequently  used 
(1  kilo- volt-ampere  = 1000  volt-amperes). 

The  reading  of  a wattmeter  applied  to  the  circuit , which  gives  the 
value  of  IE  cos  6 , gives  the  True  Power  or  Watts  expended  in 
the  circuit. 

The  ratio  of  the  tvatts  to  the  volt- amperes  in  a circuit  is  generally 
called  the  Power  factor,  as  originally  suggested  by  Fleming. 


The  power  loops  for  a circuit  are  exactly  sinusoidal,  provided 
the  original  current  and  voltage  curves  are  sinusoids.  (Figs. 
197  to  200.)  When  the  voltage  and  current  coincide  in  phase, 
i.e.  their  phases  are  Coincident,  the  average  ordinate  of  the  power 
loops  is  equal  to  one  half  of  the  maximum  ordinate,  since  the 
maximum  ordinate  is  equal  to  imem,  and  the  average  ordinate  is 


equal  to  IE  = 


2 


When  the  original  curves  are  sinusoids 


but  their  phases  are  not  coincident,  the  average  power  ordinate 
is  equal  to  one  half  of  the  difference  between  the  maximum 
value  of  the  positive  ordinate  and  the  maximum  value  of  the 
negative  ordinate.  When  the  curves  are  not  sinusoids,  the 
power  loops  are  not  sinusoidal  and  the  average  power  ordinate 
does  not  necessarily  depend  at  all  upon  the  maximum  ordinate. 

When  either  or  both  the  current  and  voltage  waves  are  not 
sinusoidal,  the  power  factor  may  be  determined  by  dividing  the 
volt-amperes,  obtained  from  amperemeter  and  voltmeter  read- 
ings made  in  the  circuit,  into  watts  obtained  from  a watt- 

p 

meter  reading;  from  which  -— = cos  d,  where  cos  6 is  a positive 

El 

fraction  less  than  unity.  Under  these  circumstances  there  is  no 
fixed  angle  of  phase  difference  which  may  be  measured  from  the 
relations  of  the  plotted  current  and  voltage  curves  and  called 

p 

the  angle  of  lag.  But  the  angle  6 = cos-1  — may  be  called  the 

IE 

equivalent  angle  of  lag.  It  is  equal  to  the  angular  displacement 


326 


ALTERNATING  CURRENTS 


between  equivalent  sinusoids* *  of  voltage  and  current  which 
produce  the  same  expenditure  of  power  in  the  circuit. 

It  must  not  be  assumed  that  curves  of  voltage  and  current 
which  are  not  sinusoidal  and  which  have  no  apparent  angular 
displacement  with  respect  to  each  other,  or  that  dissimilar  curves 
in  which  the  zero  ordinates  occur  simultaneously  will  neces- 
sarily give  power  factors  of  unity.  In  fact,  potver  factor  of 
unity  can  be  obtained  only  when  the  voltage  and  current  curves  are 
exactly  similar  and  have  no  displacement  with  reference  to  each 
other.  Curves  of  current  and  voltage  that  are  not  angularly 
displaced  with  reference  to  each  other,  but  are  not  similar,  pro- 
duce a power  factor  less  than  unity.  Curves  of  current  and 
voltage  of  relative  symmetry,  such  as  a semicircle  associated  with 
a parabola,  yield  their  maximum  power  factors  when  the  zeros  of 
current  and  voltage  occur  simultaneously.  Their  maxima  then, 
likewise,  occur  simultaneously.  When  the  current  and  voltage 
curves  are  not  of  such  relative  symmetry,  the  maxima  will  not  be 
coincident  in  time  when  the  zeros  are,  and  the  maximum  power 
factor  is  yielded  when  neither  zeros  nor  maxima  are  coincident. 

These  theorems  are  easily  proved  and  illustrated. 


Power  factor 


p 

ei~ 


But  e = e'm  sin  a + e"m  sin  2 a + e"'m  sin  3 a + etc. ; 


- C eHu  = Ef  + Ef  + Ef  + etc. ; 

7T 

i = i'm  sin  («  — 0f)+  i"m  sin  (2  a — Of)  + if"  sin  (3  a — Of)  + etc. : 

- f*  Pda  = If  + If  + If  + etc. : 

and 


1 
7 r 


eida  = E^f  cos  0X  + E2I2  cos  02  + E3I3  cos  03  + etc. 


Hence 

p.f.= 


EXIX  cos  0 j + E2I2  cos  02  + E3I3  cos  03  + etc. 

(Ef  + Ef  + Ef  + etc.)^(/j2  + If  + If  + etc.)4 


* Art.  90. 


POWER,  POWER  FACTOR 


327 


Two  conditions  must  be  fulfilled  to  make  this  unity : 

E I E T 

(1)  -1  — 1 — = 1,  etc.,  which  is  equivalent  to  saying  that 


the  voltage  curve  and  current  curve  must  be  alike  in  form  ; and 
(2)  = 0,  #2  = 0,  03=  0,  etc.,  which  is  equivalent  to  saying 

that  phases  of  corresponding  harmonics  must  be  in  coincidence 
with  each  other. 

As  an  example  of  the  association  of  two  curves  not  of  relative 


Fig.  202.  — Conventional  Curves  of  Voltage  and  Current  with  coincident  Zeros. 

symmetry,  consider  those  illustrated  in  Fig.  202,  which  shows  a 
right-angled  triangle  and  a sinusoid  with  their  zeros  coincident. 
The  formulas  of  the  curves  for  one  half  period  are 


1 * . . 

e - — and  i — im  sin  «. 


7 r 


The  effective  values  are, 


and 


The  volt-amperes  therefore  become 


The  power  is 


— sin  a — « cos  a 


•1 


7T 


328 


ALTERNATING  CURRENTS 


The  power  factor  is  p.f.  = -p-  = — - = .78.  The  two  curves 

Jhl  7 T 

shown  in  Fig.  202  therefore  give  a power  factor  of  only  78  per 
cent  when  their  zero  ordinates  coincide,  and  instrument  readings 


7T  ® 

Fig.  203.  — Conventional  Curves  displaced  — . 

in  the  circuit  would  show  an  apparent  angle  of  lag  of 
6 = cos-1  .78.  Now  suppose  the  current  curve  is  retarded  a 
quarter  cycle  as  shown  in  Fig.  203.  The  equations  become, 
for  a half  period, 


e — 


^ m a 

7T 


and  i = im  sin  ( « — ^ )• 


The  power  now  is 

P = - f « sin  (a  - p)  da 

IT  V '£J 


— — sin  a -f  cos  « 

Hence  the  power  factor  is 


1 emir 


o + if 


= -203  emim. 


p.f.  = T = .203f„;,x  ^ = .497. 

-tjl  ^rn^rn 

The  power  for  any  displacement  k of  the  zero  points  of  these 
curves  with  respect  to  each  other  may  be  expressed  thus : 


p = gin  , _ K^da 
7T“  */0 

e i Cn 

= I « ( sin  a cos  k — cos  a sin  /c)  da 

TT*  J 0 

= j^cos  k ^sin  a — a cos  aj  — sin  k ^cos  « + a sin  rcj 
= e-^f  [tt  cos  k + 2 sin  *]. 

7T-  L 


POWER,  POWER  FACTOR 


329 


The  maximum  power  factor  must  occur  when  the  power 
itself  is  a maximum.  Solving  the  above  equation  for  a maxi- 
mum gives 

^ = cos  K + 2 sin  *) 

die  <XK  7T^ 


= ^(-7rsin«  + 2cos  *). 

Equating  this  to  zero  and  simplifying  gives 

2 

— cos  k — sin  k = 0. 

7T 

Hence,  k = 32°  30'  when  the  power  to  be  derived  from  these 

p 

two  curves  is  a maximum,  and  then  — - = .377  emim  x — — = .924 

^ m i m 

and  the  angle  of  lag  indicated  by  instrument  readings  would  be 
6 — cos-1  .924  = 22°  29'.  The  maximum  power  factor  thus  be- 
comes 92.4  per  cent  for  curves  of  these  particular  forms. 

It  is  obvious  that  the  power  factor  will  be  zero  when  the  re- 
sultant power  is  zero  or  when 

P = e-^P  [tt  cos  K + 2 sin  *1  = 0, 


under  which  conditions  the  angle  k is  122°  30'  or  — 57°  30'  and 
the  angle  of  lag  indicated  by  instrument  readings  would  be 
6 = cos"1  0 = ± 90°. 

The  latter  values  of  k are  90°  different  from  its  value  when 
power  and  power  factor  are  at  their  maximum  values.  This 
obviously  must  be  so  in  this  instance  because  one  of  the  curves 
is  a pure  sinusoid,  and  therefore  maximum  power  occurs  when 
its  phase  and  the  phase  of  the  fundamental  harmonic  of  the 
other  curve  are  coincident,  and  zero  power  factor  must  occur 
when  these  two  are  in  quadrature. 

The  above  example  is  especially  convenient  for  purposes  of 
illustration  on  account  of  the  simplicity  of  the  curve  formulas  ; 
and  the  results  are  comparable  to  those  found  in  actual  electri- 
cal circuits.  However,  for  the  sake  of  closer  approximation  to 
practical  conditions,  consider  now  a peaked  curve  of  voltage  and 
a flat-topped  curve  of  current  each  composed  of  two  harmonics 
as  shown  in  Fig.  204,  and  represented  by  the  following  formulas : 

e = 9 sin  a — 3 sin  3 «, 
i — 9 sin  a + 3 sin  3 a. 


330 


ALTERNATING  CURRENTS 


The  effective  values  of  these  are 


■®=Vf+f  = 6-7and7=Aj| 


92  , y2  ft  7 

5 +T=6-7' 


The  volt-amperes  are,  therefore,  El  = 45. 


Fig.  204.  — Peaked  Voltage  and  Flat-topped  Current  Curves.  Two  Harmonics. 

Angle  k = 0. 


The  power  is 


P = - Ceida  = - f 

77*^0  77*^0 


(9  sin  a — 3 sin  3 a)  (9  sin  a + 3 sin  3 a)  da, 


-iff 

7T  L7  0 


81  sin2  ada 


- r 

•A) 


9 sin2  3 ada 


= 36. 


Or,  using  the  formulas  of  Art.  90, 

P = i COS  + 2 COS  6>g, 

= 41-  cos  0°  + a cos  180°  = 40.5  - 4.5  = 36. 


An  inspection  of  Fig.  204  affords  an  interpretation  of  this  for- 
mula. The  first  term  of  the  first  numerical  member  is  evidently 
the  power  given  by  the  primary  harmonics  which  are  of  coinci- 
dent phases,  and  the  power  loops  for  which  are  marked  Pv 
The  second  term  is  the  power  given  by  the  third  harmonics, 
which  is  shown  as  P3  in  the  power  loops,  and  is  negative  as  the 
curves  are  in  opposition  or  180°  apart.  The  power  factor  is 

p 

p.  f.  = -— =.80,  and  the  apparent  angle  of  lag  is  0 = cos-1 
El 

.80  = 36°  52'. 

In  this  instance  not  only  are  the  zeros  of  current  and  voltage 
coincident,  but  the  pairs  of  harmonics  each  have  unity  power 
factor  (positive  or  negative)  as  they  are  either  in  exact  coin- 
cidence or  exact  opposition. 


POWER,  POWER  FACTOR 


331 


If  the  current  curve  is  shifted  90°  to  the  right,  the  formula  is 
P = — cos  90°  + | cos  27 0°  = 0,  and  p.  f . = = cos  6 = 0. 


Shifting  the  fundamental  of  current  and  voltage  from  coin- 
cidence to  quadrature  generally  does  not  cause  all  the  harmonics 
to  fall  into  quadrature  as  in  this  example.  For  instance,  if  the 
third  harmonic  of  the  current  is  i,„3  sin  (3  a + 90°)  instead  of 
i sin  3 «,  the  last  formulas  above  become 

P = -M.  cos  90°  + cos  0°  =-  4.5  and  p.f.  = Fr  = cos  6 = .10. 

A x 7 7T 


In  general  the  power  factor  of  a circuit,  as  already  demon 
strated,  may  be  expressed  thus : 


cos  6 = 


where  T7 and  Tare  the  effective  voltage  and  current;  or  in  the  form 

cos  6 = F^  C0S  + F*T*  C0s  e?‘  + •••  + cos  6n 

PI 

The  following  relations  are  expressed  in  the  formulas : 

a.  The  power  factor  in  the  case  of  sinusoidal  curves  is  unity 
when  the  phases  of  voltage  and  current  are  in  coincidence  or 
opposition,  and  is  zero  when  they  are  90°  apart. 

b.  When  the  voltage  and  current  curves  are  not  sinusoids, 
the  power  factor  is  unity  when  the  curves  are  in  coincidence  or 
opposition,  provided  the  curves  are  of  exactly  like  forms.  In 
this  case,  coincidence  occurs  when  the  voltage  and  current  curves 
are  in  such  relative  phases  that  each  current  harmonic  crosses 
the  zero  value  at  the  same  instant  and  in  the  same  direction  as 
the  corresponding  voltage  harmonic  ; that  is,  each  current  har- 
monic is  in  coincidence  with  the  corresponding  voltage  har- 
monic. The  current  and  voltage  are  in  opposition  when  each 
current  harmonic  is  in  opposition  to  the  corresponding  voltage 
harmonic. 

c.  If  the  current  and  voltage  curves  are  not  of  exactly  like 
forms,  the  power  factor  never  becomes  as  large  as  unity  for  any 
phase  relation  of  the  curves;  and  it  may  pass  through  zero 
when  the  power  factors  of  some  or  all  of  the  harmonics  are  finite, 


332 


ALTERNATING  CURRENTS 


but  the  algebraic  summation  of  power  produced  by  the  several 
harmonics  is  zero. 

Power  factor  is  expressed  as  a matter  of  convenience  and 
habit  in  terms  of  the  cosine  of  an  angle  between  0°  and  180°. 
The  foregoing  discussion  shows  that  in  the  case  of  sinusoidal 
current  and  voltage,  this  angle  is  the  same  as  the  true  angle  of 
lag  or  difference  of  phase  between  the  curves.  It  is  therefore 
usual  to  call  the  cos'1  p.  f.  the  angle  of  lag,  whatever  form  may 
he  assumed  by  current  and  voltage  curves.  When  the  current 
and  voltage  are  sinusoidal,  the  angle  of  lag  may  he  scaled  off 
from  a plot  of  the  curves,  hut  this  cannot  readily  be  done  when 
the  curves  are  of  other  forms  on  account  of  the  effects  of  the 
various  harmonics. 

When  the  power  factor  is  positive , the  circuit  is  absorbing  poiver 
from  an  outside  source  ; when  it  is  negative , the  circuit  is  deliver- 
ing power  to  the  source  of  the  voltage  under  consideration. 

Proh.  1.  A circuit  has  an  effective  sinusoidal  voltage  of  500 
volts  impressed  upon  it  and  a sinusoidal  current  of  75  amperes 
flows  at  a lag  of  30°.  What  are  the  volt-amperes  of  the  circuit, 
the  power  expended,  and  the  power  factor  ? 

Proh.  2.  Two  circuits  are  in  series,  one  having  a resistance 
of  5 ohms  and  a self-inductance  of  .01  of  a henry,  the  other  a 
resistance  of  10  ohms  and  a capacity  of  200  microfarads,  and  a 
sinusoidal  current  of  50  amperes  flows  through  these  two  circuits 
at  a frequency  of  40  periods  per  second.  What  is  the  power  fac- 
tor of  the  combined  circuits  and  of  each  circuit  separately  ? 

Prob.  3.  A circuit  is  composed  of  four  circuits  in  parallel, 
the  first  having  a resistance  of  20  ohms,  the  second  a resistance 
of  5 ohms  and  a self-inductance  of  .01  of  a henry,  the  third  a 
resistance  of  8 ohms  and  a capacity  of  100  microfarads,  the 
fourth  a resistance  of  2 ohms,  a self-inductance  of  .02  of  a henry, 
and  a capacity  of  200  microfarads.  When  1000  volts  (sinu- 
soidal) are  impressed  upon  this  circuit  at  a frequency  of  60 
periods  per  second,  what  is  the  power  factor  in  the  entire  cir- 
cuit, and  of  each  of  the  four  parts?  Give  the  angle  of  lag  or 
lead  in  each  part  of  the  circuit  and  in  the  total  circuit. 

Proh.  4.  A voltage  having  two  harmonics,  the  first  and  the 
third,  of  which  the  effective  values  are  respectively  200  volts 
and  10  volts,  is  impressed  upon  a circuit  and  a current  flows 
having  first  and  third  harmonics  with  effective  values  of  50 


POWER,  POWER  FACTOR 


B33 


and  5 each  in  coincidence  with  the  corresponding  voltage  har- 
monic. What  is  the  power  factor? 

Prob.  5.  A voltage  having  two  harmonics,  the  first  and  the 
third,  with  effective  values  of  200  and  50,  is  impressed  at  a fre- 
quency of  60  periods  per  second  on  a circuit  having  a capacity 
of  200  microfarads  and  a resistance  of  5 ohms  in  series.  What 
current  flows  in  the  circuit  and  what  is  the  power  factor? 

Prob.  6.  The  voltage  of  problem  5 is  impressed  upon  a cir- 
cuit having  a resistance  of  10  ohms  and  a self-inductance  of  .01 
of  a henry  in  series.  What  is  the  power  factor  ? 

Prob.  7.  The  voltage  of  problem  5 is  impressed  on  a circuit 
having  5 ohms  resistance,  200  microfarads  capacity,  and  .2  of  a 
henry  self-inductance  in  series.  What  is  the  power  factor? 

Prob  8.  The  voltage  of  problem  5 is  impressed  upon  a cir- 
cuit having  a capacity  of  200  microfarads.  What  is  the  power 
factor?  What  would  be  the  power  factor  if  a sinusoidal  vol- 
tage of  the  same  effective  value  were  impressed  on  this  circuit  ? 

Prob.  9.  The  voltage  of  problem  5 is  impressed  upon  a cir- 
cuit having  a self-inductance  of  .2  of  a henry.  What  is  the 
power  factor?  What  would  be  the  power  factor  if  a sinusoi- 
dal voltage  of  the  same  effective  value  were  impressed  on  the 
circuit? 

Prob.  10.  The  voltage  of  problem  5 is  impressed  upon  a cir- 
cuit having  5 ohms  resistance.  What  is  the  power  factor? 
What  would  be  the  power  factor  if  a sinusoidal  voltage  of  the 
same  effective  value  were  impressed  on  the  circuit? 

Prob.  11.  What  would  be  the  power  factor  in  problem  4 if 
the  third  harmonic  of  current  was  in  opposition  to  the  third 
harmonic  of  the  voltage  ? 

Prob.  12.  A voltage  curve  having  loops  in  the  form  of  a 
semicircle  and  with  a maximum  value  of  200  volts  is  in  coinci- 
dence with  a sinusoidal  current  having  a maximum  value  of 
100  amperes.  What  is  the  power  factor? 

95.  Expression  of  Power  Relations  by  Means  of  Vectors.  — 

The  product  of  the  vectors  of  current  and  voltage  may  be  called 
Vector  power.  Apparent  power,  which  is  given  by  the  product 
of  the  readings  of  the  amperemeter  and  voltmeter  in  the  circuit, 


334 


ALTERNATING  CURRENTS 


or  the  “ volt-amperes,”  is  numerically  equal  to  the  product  of  the 
tensors  of  current  and  voltage,  and  is  a coefficient  in  vector 
power.  Apparent  power  may  be  resolved  into  two  components 
in  quadrature  about  the  argument  8.  One  of  these  components 
is  equal  to  IE  cos  6 , or  the  reading  of  a wattmeter  in  circuit,  and 
is  called  real  power  or  true  power.  The  component  perpendicu- 
lar thereto  is  equal  to  jlE  sin  d,  and  is  the  power  which  is  exerted 
alternately  positively  and  negatively  in  the  circuit,  but  which 
balances  itself  in  and  out  during  any  half  period  so  that  its 
average  is  equal  to  zero.  The  existence  of  the  second  compo- 
nent is  dependent  on  a condition  that  during  one  portion  of  the 
half  period  the  circuit  absorbs  power  and  during  the  remaining 
portion  the  circuit  delivers  power  to  an  exactly  equal  average 
amount,  so  that  a wattmeter  reading  will  be  zero  with  respect 
to  this  latter  component.  Hence  comes  the  absurdly  inappli- 
cable term  wattless  energy  or  power,  which  can  better  be  called 
Quadrature  power  or  Reactive  volt-amperes.  It  may  also  be 
called  the  quadrature  volt-amperes.  A mechanical  analogy  of 
the  reactions  of  the  quadrature  power  during  a half  period  is 
found  in  a cycle  consisting  of  the  frictionless  raising  of  a weight 
and  its  frictionless  return  to  its  initial  position.  In  this  case 
mechanical  power  is  exerted  by  some  source  to  raise  the  weight, 
but  the  weight  delivers  back  the  full  power  to  the  source  as  it 
returns  to  its  first  position.  The  weight  has  here  had  work 
exerted  on  it  by  the  source'  and  has  then  returned  an  equal 
amount  of  work  to  the  source,  and  a mechanical  power  meter 
would  afford  a zero  reading  as  the  result  of  this  operation. 

The  power  vector  has  the  form 

P = Pcjs-«/r, 

and  it  is  equal  to  the  product  of  the  current  and  voltage  vectors, 
or  P = El  = E(  cos  a+j  sin  oc)  • Z[cos(«  — 8)  + j sin  (a  — 0)] 
= Ee]a  x Iej(a~8)  = IEe3<2a~9)  = IEcjs  (2  a — 8) 

= (IE  cos  8 — jlE  sin  0)cjs  2 a. 

Cjs-v/r  is  therefore  equal  to  cjs  2 a,  and  it  is  clearly  seen  that 
P has  twice  the  frequency  of  Zand  Z7,  as  heretofore  explained. 
Moreover,  the  apparent  power  P is  shown  to  be  composed 
of  two  rectangular  components.  One  of  these  (IE  cos  9 ) 
represents  the  true  average  power  in  the  period,  and  the  other 
(jlE  sin  d)  is  the  quadrature  power  which  disappears  when 


POWER,  POWER  FACTOR 


335 

taken  over  any  complete  half  period  and  stands  for  the  work 
that  swashes  back  and  forth  in  the  circuit  but  always  gives  as 
much  as  it  takes  in  any  complete  half  period. 

By  drawing  a curve  through  the  points  found  by  giving 
various  values  to  « from  0°  to  360°  in  the  expression  Pcjs  2 a,  we 
get  the  well-known  sinusoidal  “power  loops  ” which  are  of  twice 
the  frequency  of  the  corresponding  vectors  of  voltage  and  current. 
The  true  power  is  zero  when 

cos  6=0, 

that  is,  when  6 = ± 90°,  in  which  case  E and  I are  in  quadra- 
ture; and  the  quadrature  power  is  zero  when 

sin  6 = 0, 

that  is,  when  6 = 0°  or  180°,  in  which  case  E and  I are  either  in 
coincidence  or  in  opposition. 

The  algebraic  sign  of  the  angle  6 indicates  whether  the 
quadrature  power  is  the  result  of  equivalent  inductance  or 
equivalent  capacity;  and  the  numerical  value  of  6 indicates 
whether  the  true  power  is  delivered  or  absorbed  by  the  circuit, 
power  being  absorbed  by  the  circuit  when  ± 6 < 90°,  and  de- 
livered by  the  circuit  when  ± 6 > 90°.  The  application  of  the 
commutative,  associative,  and  distributive  laws  of  algebra  to 
the  vector  formulas  is  not  affected  by  doubling  the  frequency 
in  the  equation. 

These  same  deductions  can  be  made,  using  the  ordinary  com- 
plex form  of  the  vectors.  In  this  case  the  product  of  current 
and  voltage  gives : 

El  = (a  +jb')(^c  +jd')  = ac  — bd  + j ( ad  + bc~) 

= El  j [cos  « cos  («  — d)  — sin  « sin  (a  — d)] 

+ j [sin  a cos  (a  — d)  + cos  « sin  («  — d)]  j 
= -ET[cos(2  a — d)  +j  sin  (2  « — d)]. 

This  expression  is  the  complex  expression  in  which  are  shown 
the  horizontal  and  vertical  components  of  the  complete  power 
vector;  that  is, 

ac  — bd  = El  cos  (2  a — d) 
and  ad  + be  = El  sin  (2  a — d). 

This  is  not  what  we  desire,  as  the  valuable  information  to  be 
derived  from  the  equation  is  the  relative  magnitude  of  the  rec- 


336 


ALTERNATING  CURRENTS 


tangular  components  of  the  apparent  power  corresponding  to  the 
argument  0,  since  the  average  of  the  variable  angle  « over  a 
period  is  zero  and  0 is  constant.  The  forms  given  earlier  have 
shown  that  apparent  power  is  a coefficient  in  the  vector  power 
and  is  multiplied  by  (cos  6 — j sin  d)cjs  2 « to  give  the  vector 
power.  We  now  desire  to  transform  El  (cos  0 —j  sin  0)  into 
terms  of  the  rectangular  components  of  voltage  and  current  a,  b, 
c , and  d.  But, 

cos  0 = cos  [«  — («  — #)]  (see  Fig.  205) 

= cos  a cos  (a  — 0)  + sin  a sin  (a  — 0) 
a c , b d _ac-\-bd 
~e1+E'  I ~ El 

and,  therefore, 

El  cos  0 = ac  4-  bd. 

Also,  sin  0 = sin  [«  — (a  — #)] 

= sin  a cos  (a  — 0)  — cos  « sin  (a  — 0s) 

_ b c a d __bc  — ad 

~E  ' I~E  ' I~  El  ’ 

and,  therefore, 

jEI  sin  0 =j(bc  — ad). 

The  true  power  is  therefore  equal  to  ac  + bd  and  not  to  ac—hd\ 
and  the  quadrature  power  to  be  — ad  and  not  to  be  -f  ad. 

It  is  to  be  remembered 
that  the  true  power,  ac + 
bd,  is  the  average  value 
for  any  full  period  of  a 
periodic  quantity  of  double 
frequency,  Pcjs  2 a,  the 
origin  of  which  is  displaced 
from  the  W-axis  by  a dis- 
tance equal  to  El  cos  0 ; 
and  the  quadrature  power 
be  — ad  is  the  quadrature 
component  of  El  obtained  by  subtracting  (ac  -p  bd )2  from 
(Eiy. 

The  conditions  are  illustrated  in  Fig.  205,  in  which  OA  and 
OB,  respectively,  represent  vector  voltage  and  current,  OC  rep- 
resents vector  power,  and  OB  and  BO,  respectively,  represent 
IE  cos  0 = ac  + bd  and  IE  sin  0=  be  — ad ; OE  and  EC  repre- 


c 


Fig.  205.  — Vector  Diagram  of  Power  Relations. 


POWER,  POWER  FACTOR 


337 


sent  the  horizontal  and  vertical  components  of  vector  power 
obtained  by  the  multiplication  of  the  complex  quantities,  i.e. 
ac  — bd  and  ad  + be. 

The  expression  -Z/i  (cos  6 —j  sin  0)  is  of  the  nature  of  a vector 


operator,  similar  in  character  to 

— 7^2 

El  (cos  0 —j  sin  0)  = E2Y  = — • 

z 


r = 


Jr  (cos  0-j  sin  0)  ; 


and 


Similar  deductions  are  reached  when  the  expressions  are  still 
further  generalized  by  assigning  an  initial  fixed  angle  to  the  vol- 
tage vector,  in  which  case  E = Ecjs  (a  — </>)and  I = 7ejs(a  — <p  — 0). 

Since  the  power  factor  is  equal  to  true  power  divided  by  volt- 
amperes,  it  must  be  equal  to  cos  0.  In  the  same  way  the  in- 
ductance factor  which  is  equal  to  quadrature  power  divided  by 
volt-amperes  must  be  equal  to  sin  0. 

We  see  from  what  goes  before  that  the  volt-amperes  of  a com- 
pound circuit  may  not  be  equal  to  the  algebraic  sum  of  the  volt- 
amperes  in  the  parts  of  the  circuit,  but  they  must  always  be  equal 
to  the  vector  sum  of  the  volt-amperes  in  the  parts  of  the  circuit. 

Inasmuch  as  true  power  and  quadrature  power  in  the  several 
parts  of  a circuit  are  the  rectangular  components  of  the  respec- 
tive volt-amperes  taken  with  respect  to  the  corresponding  argu- 
ments 0V  0V  etc.,  it  is  clear  that  the  true  power  in  any  circuit 
is  equal  to  the  algebraic  sum  of  the  powers  in  the  parts  of  the 
circuit,  and  the  quadrature  power  in  the  total  circuit  equals  the 
algebraic  sum  of  the  quadrature  power  in  the  parts. 

The  conditions  that  are  here  set  forth  hold  regardless  of 
whether  the  parts  of  the  circuit  are  in  series  or  in  parallel. 

Power  will  be  assumed  to  be  positive  in  every  case  where 
it  is  absorbed  by  a circuit  and  negative  when  it  is  delivered 
from  the  same. 

Considering  the  semicircle  diagrams  of  Figs.  142  and  143, 
the  impressed  voltage  is  supposed  to  be  constant  and  drawn 
vertically  upwards,  and  the  vertical  component  of  current  in 
each  instance  is  the  active  component.  Then  the  power  ex- 
pended in  the  circuit  is  equal  to  the  voltage  multiplied  by  the 
vertical  component  of  current.  Consequently,  the  vertical  com- 
ponent of  the  current  vector  for  each  position  in  the  semicircle 
diagram  is  proportional  to  true  power,  since  voltage  is  assumed 
to  be  constant ; and  the  power  corresponding  to  each  current 


338 


ALTERNATING  CURRENTS 


Fig.  206.  — Locus  of  the  Power  Vector  in  a Series 
Circuit  where  Resistance  varies  and  Reactance 
is  maintained  Constant. 


vector  may  be  laid  off  on  that  vector  of  a length  from  0 which  is 
proportional  to  the  vertical  component.  Following  this  process 

for  the  Figs.  142  and 
143,  gives  Figs.  206  and 
207,  in  which  the  power 
vector  locus  is  repre- 
sented by  an  oval-like 
loop  within  each  semi- 
circle. The  power  per- 
taining to  any  current 
vector  is  equal  to  the  in- 
tercept on  that  current 
between  0 and  the  oval 
cutting  it,  as,  for  in- 
stance, the  power  corre- 
sponding to  the  current 
01  with  the  given  voltage  is  represented  by  OP , which  is 
numerically  equal  to  the  vertical  component  of  current  01  when 
P is  unity.  It  will  be  observed  that  the  ovals  do  not  give  any 
new  information  about  the  nu- 
merical value  of  the  power  for 
each  current  vector,  other  than 
that  to  be  obtained  directly  from 
the  active  current  components 
in  the  semicircle  diagrams,  but 
the  ovals  show  in  a graphic 
manner  the  way  in  which  the 
power  varies  with  the  current 
over  the  entire  range  of  condi- 
tions considered. 

It  will  be  observed  that  each 
semicircle  in  Fig.  206  contains 
its  own  oval-like  loop.  Of  the 
four  ovals  occurring  in  this  fig- 
ure, two  are  positive  and  two 
are  negative  as  indicated.  The 
positive  ovals  represent  condi- 
tions in  which  power  is  absorbed 

by  and  expended  in  the  circuit  by  conversion  into  heat,  light, 
or  mechanical  power ; while  the  negative  ovals  represent  the 


Fig.  207.  — Locus  of  the  Power  Vector  in 
a Series  Circuit  where  Reactance  varies 
and  Resistance  is  Constant. 


POWER,  POWER  FACTOR 


339 


conditions  when  the  circuit  generates  and  gives  up  energy  as 
in  the  case  of  an  electrical  generator.  In  each  of  these  figures 
the  semicircles  to  the  right  of  the  vertical  axis  represent  condi- 
tions when  the  current  lags  behind  the  voltage,  and  the  other 
semicircles  represent  conditions  when  the  current  leads  the 
voltage.  In  passing  from  one  oval  to  another  in  either  figure, 
it  will  be  observed  that  either  the  current  or  the  power  factor 
passes  through  zero. 

It  will  be  observed  that  the  ovals  of  Fig.  206  show  the  maxi- 
mum value  of  power  when  the  angle  of  lag  in  the  circuit  is  either 
± 45°,  or  ± 135°.  It  is  easy  to  prove  that  the  condition  of  maxi- 
mum possible  power  transferred  through  an  alternating-current 
circuit  with  a fixed  voltage  and  with  fixed  reactance  and  variable 
resistance,  as  illustrated  by  Fig.  206,  comes  when  the  angle  of 
lag  is  ± 45°  or  ± 135°.  The  demonstration  is  as  follows : 

R E 

Since  P = IE  cos  6 and  since  cos  9 = — and  1=  — , there  re- 
sults the  equation: 


P = 


E2R 
R2  + X2' 


This  is  obviously  a maximum  when  R = X,  at  which  time  6 = 45° 
and  the  power  factor  is  70.7  per  cent.  It  is  equally  easy  to 
prove  that  the  condition  of  maximum  possible  power  transferred 
through  an  alternating  current  circuit  with  a fixed  voltage  and 
with  fixed  resistance  and  variable  reactance,  as  illustrated  in 
Fig.  207,  comes  when  the  angle  of  lag  is  0°  or  180°.  In  this 
case,  as  before, 

E2R 


P = 


R2  + X2' 


whicli  is  obviously  a maximum  when  X=  0.  Under  these  cir- 
cumstances the  current  is  in  coincidence  with  the  voltage 
(0  = 0°)  when  the  circuit  absorbs  power,  and  in  opposition  to 
the  voltage  (6  = 180°)  when  the  circuit  delivers  power. 

It  may  be  noted  in  this  connection  that  the  condition  of  maxi- 
mum power  is  a condition  of  little  practical  importance,  as  the 
highest  practicable  operating  efficiency  or  plant  efficiency  is 
usually  desired  in  the  operation  of  electrical  circuits  and 
machinery.  A high  plant  efficiency  is  ordinarily  incompatible 
with  a low  power  factor. 


340 


ALTERNATING  CURRENTS 


96.  Quadrature  Components  of  Power  Loops.  — If  6 , the  angle 
of  lag  in  a circuit,  were  ± 90°,  the  total  current  would  be  in 
quadrature  with  the  voltage,  and  therefore  its  vector  product 
with  the  voltage  would  be  zero,  as  already  indicated.  A quad- 
rature current  may  do  a considerable  amount  of  work  in  one 

J,I»r 

sin  a sin  («—  90°)rfa  = J,but  during  the 
o 

next  quarter  period  the  circuit  returns  an  equal  amount,  and 
the  total  work  for  the  period  is  zero.  (Compare  power  loops, 
Fig.  199.)  It  is  possible  to  make  the  ratio  of  inductive  reac- 
tance, 2 7r/X,  so  great  in  comparison  with  the  true  resistance 
R of  a circuit  that  the  lag  is  nearly  90°.  It  is  also  possible  to 
make  the  capacity  of  a circuit  so  great  in  comparison  with  its 
resistance  that  there  is  a lead  of  nearly  90°.  The  latter  condi 
tion  is  one  not  ordinarily  met  in  practice,  but  the  former  may 
quite  easily  be  brought  about  in  circuits  including  underloaded 
transformers  of  poor  design. 

The  effect  of  the  quadrature  element  of  a current  lagging 
less  than  90°  may  be  shown  by  components  as  indicated  in  Fig. 

208,  where  the  full  line 
curves  E,  I,  IA,  and  Ig, 
respectively,  represent 
the  voltage,  current,  ac- 
tive component  of  cur- 
rent, and  the  quadrature 
component  of  current, 
while  the  dash-dot  line 
represents  the  power 
loops  and  the  dotted 
lines  the  components  of 
the  power  loops  pro- 
vided respectively  by  the 
active  and  quadrature 
components  of  current. 
It  is  here  seen  that  during  each  half  period  equal  amounts  of 
power  are  received  and  given  out  on  account  of  the  quadrature 
component,  while  the  active  component  causes  energ}'  to  be 
absorbed  by  the  circuit  throughout  the  period.  The  power 
loops  due  to  the  quadrature  component  really  represent  the 
work  stored  in  the  magnetic  field  as  the  current  rises,  or  stored 


Fig.  208.  — Power  Loops  caused  "by  Active  and 
Quadrature  Components  of  the  Current  I.  Lag 
45°. 


POWER,  POWER  FACTOR 


341 


in  the  electrostatic  field  as  the  voltage  rises,  and  given  out  as 
it  falls. 

Figure  209  represents  the  component  power  loops  formed  by 
the  product  of  the  current  respectively  with  the  active  and 
quadrature  compo- 
nents of  the  voltage. 

The  total  current,  to- 
tal voltage,  and  total 
power  curves  are  iden- 
tical with  those  of 
Fig.  208. 

The  power  loops 
representing  total  in- 
stantaneous power  are 
sinusoidal  when  E and 
I are  sinusoids,  and 
they  Vary  in  position  ^IG-  — Power  Loops  caused  by  the  Active  and 

. . . Quadrature  Components  of  the  Voltage  E,  Lag  45°. 

with  respect  to  the 

X-axis  from  just  touching  that  axis  when  6 = 0°  to  a symmetri- 
cal location  thereon  when  6 = 90°.  The  power  loops  due  to 
the  active  component  of  current  or  voltage  represent  the  irre- 
versible power  (the  average  of  which  over  each  period  we  call 
true  power)  and  are  always  located  so  as  to  just  touch  the  X- 
axis  ; while  the  loops  due  to  the  quadrature  component  of 
current  or  voltage  represent  the  self-compensating  quadrature 
or  reactive  power  and  are  always  located  symmetrically  on  the 
X-axis  with  equal  areas  above  and  below  the  axis. 

Prob.  1.  A circuit  has  a reactance  of  10  ohms  and  a resist- 
ance of  10  ohms.  When  a voltage  of  2000  volts,  at  a particular 
frequency  required  to  give  the  reactance  named,  is  impressed 
upon  it,  what  are  the  values  of  the  active  and  quadrature  com- 
ponents of  the  current  ? Work  this  and  the  following  problems 
by  means  of  complex  quantities. 

Prob.  2.  A total  current  of  50  amperes  at  the  given  fre- 
quency flows  through  the  circuit  in  problem  1.  What  is  the 
impressed  voltage,  and  what  are  the  values  of  the  active  and 
quadrature  components  of  the  current  ? 

Prob.  3.  A circuit  composed  of  three  parts  in  series,  the  first 
having  a resistance  of  10  ohms,  the  second  a self-inductance  of 


342 


ALTERNATING  CURRENTS 


.02  of  a henry,  and  the  third  a capacity  of  100  microfarads,  has 
a sinusoidal  voltage  of  500  volts  with  a frequency  of  120  periods 
per  second  impressed  upon  it.  What  are  the  values  of  the  active 
and  quadrature  currents  in  the  total  circuit  and  in  each  part 
taken  respectively  with  reference,  (1)  to  the  voltage  on  the 
total  circuit,  and  (2)  to  that  on  the  particular  part  considered  ? 

97.  Reactive  Factor;  Table  of  Reactive  and  Power  Factors. 

— The  value  of  the  power  factor  of  a circuit  is  equal  numeri- 
cally to  cos  0,  and  the  total  current  in  a circuit  multiplied  by 

TABLE  OF  POWER  FACTORS  AND  REACTIVE  FACTORS 


Lag 

Power 

Reactive 

Factor 

Lag 

Power 

Reactive 
Factor 
sin  6 
± 

Angle  6 

Factor 

Angle  6 

Factor 

± 

cos  6 

± 

± 

cos  8 

Degrees 

Degrees 

Degrees 

Degrees 

0 

1.0000 

.0000 

90 

23 

.9205 

.3907 

67 

1 

.9998 

•0175- 

89 

24 

.9135 

.4067 

66 

o 

.9994 

.0349 

88 

25 

.9063 

.4226 

65 

3 

.9986 

.0523 

87 

26 

.89S8 

.4384 

64 

4 

.9976 

.0698 

86 

27 

.8910 

.4540 

63 

5 

.9962 

.0872 

85 

28 

.8829 

.4695 

62 

6 

.9945 

.1045 

84 

29 

.8746 

.4848 

61 

7 

.9925 

.1219 

83 

30 

.8660 

.5000 

60 

8 

.9903 

.1392 

82 

31 

.8572 

.5150 

59 

9 

.9877 

.1564 

81 

32 

.8480 

.5299 

58 

10 

.9848 

.1736 

80 

33 

.8387 

.5446 

57 

11 

.9816 

.1908 

79 

34 

.8290 

.5592 

56 

12 

.9781 

.2079 

78 

35 

.8191 

.5736 

55 

13 

.9744 

.2249 

77 

36 

.8090 

.5878 

54 

14 

.9703 

.2419 

76 

37 

.7986 

.6018 

53 

15 

.9659 

.2588 

75 

38 

.7880 

.6156 

52 

16 

.9613 

.2756 

74 

39 

.7771 

.6293 

51 

17 

.9563 

.2924 

73 

40 

.7660 

.6428 

50 

18 

.9511 

.3090 

72 

41 

.7547 

.6561 

49 

19 

.9455 

.3256 

71 

42 

.7431 

.6691 

4S 

20 

.9397 

.3420 

70 

43 

.7313 

.6S20 

47 

21 

.9336 

.3584 

69 

44 

.7193 

.6946 

46 

22 

.9272 

.3746 

68 

45 

.7071 

.7071 

45 

i 

sin  6 
Reactive 
Factor 

cos  6 
Power 
Factor 

± 

e 

Lag 

Angle 

± 

sin  6 
Reactive 
Factor 

cos  6 
Power 
Factor 

± 

8 

La» 

Angle 

POWER,  POWER  FACTOR 


S43 

the  power  factor  is,  therefore,  equal  to  the  active  component  of 
the  current.  A factor  which  in  the  same  way  is  proportional 
to  the  quadrature  current,  may  be  called  the  Reactive  factor  of 
a circuit.  It  is  evidently  equal  in  numerical  value  to  sin  6. 

Any  table  which,  like  the  foregoing,  gives  natural  sines  and 
cosines  of  the  circular  angles  also  gives  the  numerical  values 
of  the  reactive  factors  and  power  factors  of  the  angles. 

This  table  is  given  for  angles  of  lag;  that  is,  the  values  of  6 
are  positive.  When  the  current  leads,  that  is,  6 is  negative,  the 
numerical  value  of  the  power  factor  (cos  O')  corresponding  with 
the  particular  value  of  6 is  unchanged,  but  reactive  factor 
(sin  6)  changes  from  positive  to  negative  value.  The  reactive 
factor  lias  sometimes  been  called  the  Induction  Factor,  especially 
in  inductive  circuits. 

Prob.  1.  What  is  the  reactive  factor  in  a circuit  having  a 
resistance  of  10  ohms,  a capacity  reactance  of  10  ohms,  and  a 
self-inductive  reactance  of  10  ohms  in  series,  when  a voltage  of 
100  volts  at  a frequency  of  120  periods  per  second  is  impressed? 

Prob.  2.  The  volt-amperes  in  a circuit  equal  10,000  and  the 
reactive  factor  is  .9.  What  is  the  power? 

98.  Method  of  Measuring  Angle  of  Lag  and  Power  Factor.  — 

The  power  factor,  or  cos  9 , whether  the  curves  of  voltage  and 
current  are  sinusoidal  or  not,  may  be  determined  by  using  vol- 
tage, current, and  power  readings  taken  simultaneously, as  already 
explained.*  The  method  for  determining  the  angle  of  lag  is  as 
follows  : Measure  the  current  flowing  in  a circuit  by  a correct 
amperemeter  ; measure  the  voltage  at  its  terminals  by  an  elec- 
trostatic voltmeter  or  some  type  of  non-inductive  voltmeter  of 
relatively  very  high  resistance.  Finally,  measure  the  power 
absorbed  in  the  circuit  by  means  of  a wattmeter,  the  voltage  coil 
of  which  is  non-reactive  and  of  relatively  high  resistance.  The 
power  in  watts  determined  by  the  wattmeter,  when  divided  by 
the  product  of  volts  and  amperes,  gives  the  cosine  of  the  angle 
of  lag.  If  the  curves  of  current  and  voltage  are  of  irregular 
form,  this  measurement  will  give  the  power  factor  which  is 
equal  to  the  angle  of  lag  between  the  equivalent  sine  curves  for 
the  particular  condition  of  the  circuit.f  We  will  later  take 


*Art.  94. 


tArt.  90. 


344 


ALTERNATING  CURRENTS 


up  the  effect  of  reactance  in  the  voltage  coil  of  the  watt- 
meter.* 

Prob.  1 . The  no-load  terminal  voltage  of  a generator  having 
a frequency  of  40  periods  per  second  is  2200  volts,  as  shown  by 
a voltmeter.  When  the  machine  is  fully  loaded  on  a non- 
inductive  load,  the  voltmeter  measures  2000  volts,  the  ampere- 
meter measures  100  amperes,  and  the  wattmeter  measures  200 
kilowatts.  The  resistance  of  the  armature  is  1 ohm.  What 
is  the  angle  of  lag  between  the  induced  voltage  (2200  volts) 
and  the  current  when  the  machine  is  fully  loaded?  Is  the 
wattmeter  reading  needed  in  this  instance  ? Why  ? 

99.  Methods  for  Measuring  the  Power  in  an  Alternating-current 
Circuit. — It  can  be  readily  understood  that  the  power  in  an 
alternating-current  circuit  may  be  measured  most  accurately 
and  expeditiously  by  means  of  a wattmeter.  The  other  methods 
here  given  may  serve  a purpose  in  special  cases  or  when  a watt- 
meter of  the  proper  range  is  not  at  hand.  They  are  all  well 
worth  studying  as  problems. 

1.  Wattmeter  Methods.  Any  instrument  which  directl}r 
measures  the  true  power  in  a circuit  is  called  a wattmeter. 
The  common  form  of  a wattmeter  is  an  electrodynamometer f 
with  one  of  its  coils  connected  across  the  terminals  of  the  circuit 
under  test  and  the  other  in  series  therewith.  Such  instruments 
of  modern  form  are  usually  encased  in  a suitable  box  to  make 
them  portable,  and  are  proportioned  so  as  to  give  direct  read- 
ings. The  electrodynamometer  in  its  ordinary  amperemeter 
arrangement  with  the  two  coils  in  series  measures  the  value  of 


- Pdt,  since  the  torque  exerted  on  the  movable  coil  at  each 

instant  is  proportional  to  the  product  of  the  currents  in  the 
movable  arid  fixed  coils.  The  average  torque  is  therefore  pro- 


portional to  1 2,  since  ■ fidt,  as  has  been  earlier  shown,  is 


equal  to  1 2 ; and  the  deflection  of  the  movable  coil,  if  its  motion 
is  opposed  by  a spring,  is  D = KT2.  when  iThas  a value  fixed  by 
the  conformation  and  character  of  the  parts  of  the  instrument. 
To  make  such  an  instrument  direct  reading,  it  must  be  provided 


* Art.  99,  1 a. 


t Art.  24. 


POWER,  POWER  FACTOR 


345 


with  a square-root  scale,  and  the  scale  consequently  is  quite 
open  at  the  higher  portion,  but  contracts  badly  toward  the  lower 
parts,  because  the  units  of  graduation  are  proportional  to  the 
square  root  of  the  torque.  As  such  a scale  is  inconvenient  for 
use,  these  instruments,  when  arranged  for  use  as  voltmeters  or 
amperemeters,  are  sometimes  constructed  so  that  the  conductors 
of  the  movable  coil  are  located  in  the  densest  portions  of  the 
magnetic  field  of  the  fixed  coil  when  the  deflections  are  small. 
This  arrangement  manifestly  counteracts  in  some  degree  the 
contraction  in  the  lower  part  that  is  inherent  in  the  pure 
square-root  scale. 

The  assumption  that  the  torque  on  the  movable  coil  of  any 
such  instrument  is  directly  proportional  to  the  product  of  the 
currents  in  the  two  coils  may  be  vitiated  by  various  disturbing 
influences,  — particularly  if  the  movable  coil  in  its  deflection 
moves  so  as  to  appreciably  alter  the  relations  to  each  other  of 
the  magnetic  fields  of  the  two  coils,  and  if  eddy  currents  can  be 
set  up  in  any  metal  parts  of  the  instrument  in  such  a manner 
as  to  introduce  an  additional  and  extraneous  magnetic  field. 
If  no  such  disturbing  influences  enter  the  situation,  the  calibra- 
tion of  such  an  instrument  by  means  of  currents  of  one  fre- 
quency, or  even  by  means  of  direct  currents,  is  sufficient  to 
standardize  it  for  use  with  currents  of  any  ordinary  frequency  ; 
provided,  however,  that  if  one  or  both  of  the  instrument  coils 
are  to  be  used  across  the  line  as  voltage  coils  they  must  be 
non-reactive,  or  the  calibration  will  be  affected  by  frequency. 

An  additional  explanation  of  these  conditions  is  contained 
in  Art.  24. 

When  an  electrodynamometer  is  arranged  for  use  as  a watt- 
meter, one  coil  (ordinarily  the  movable  coil)  is  planned  for 
connection  between  the  terminals  of  the  circuit  within  which 
the  power  is  to  be  measured  and  may  be  called  the  Voltage  coil, 
and  the  other  coil  is  planned  to  be  connected  in  series  with  the 
circuit  and  may  be  called  the  Current  coil.  The  connections 
are  illustrated  in  Fig.  51.  In  this  case  the  torque  on  the  mov- 
ing coil,  at  each  instant,  is  still  proportional  to  the  product  of 
the  corresponding  instantaneous  currents  in  the  two  coils,  and 


i rT.. 

the  average  torque  is  therefore  proportional  to  — n^t,  where 


i represents  the  instantaneous  current  in  the  current  coil  and  q 


346 


ALTERNATING  CURRENTS 


the  corresponding  instantaneous  current  in  the  voltage  coil.  If 
the  coil  is  non-reactive  or  the  frequency  is  constant,  the  current 
in  the  voltage  coil  is  directly  proportional  to  the  voltage  im- 
pressed upon  the  coil,  which  is  the  same  as  the  voltage  impressed 
on  the  main  circuit,  and  therefore  il  is  proportional  to  e , where 
e is  the  instantaneous  impressed  voltage.  Hence  the  average 
torque  acting  on  the  movable  coil  of  the  wattmeter  is  propor- 
1 CT . 

tional  to—  I iedt  = P ; or,  D = KP , if  the  motion  of  the  coil  is 
27«/o 

opposed  by  a spring  and  its  deflection  is  represented  by  D.  In 
this  case,  P is  the  power  expended  in  the  circuit  to  which  the 
wattmeter  is  connected  and  K is  a constant  of  the  instrument 
with  a value  fixed  by  the  conformation  and  character  of  the 
parts  of  the  device.  It  will  be  observed  that  the  deflection  is 
here  proportional  to  the  first  power  of  P,  and  a uniformly 
graduated  scale  is  therefore  used  on  such  an  instrument  except 
as  modifications  are  required  to  correct  for  errors  of  construc- 
tion of  the  parts.  The  torque  of  a wattmeter  of  this  type, 
where  the  coils  are  non-reactive,  is  therefore  directly  propor- 
tional to  the  power,  while  the  torque  of  instruments  of  the 
same  type  used  with  the  coils  in  series  is  proportional  to  the 
square  of  the  effective  current.  In  the  usual  arrangement, 
shown  in  Fig.  51,  wattmeters  of  this  class  have  a current  coil  of 
a few  turns  of  thick  wire,  which  is  placed  in  series  with  the  cir- 
cuit to  be  measured.  The  voltage  coil  is  composed  of  a few 
turns  of  fine  wire  in  series  with  non-reactive  resistance,  and 
is  connected  across  the  terminals  of  the  circuit. 

a.  Errors  in  Wattmeter  Readings  on  Account  of  the.  React- 
ance of  the  Coils.  It  is  essential  that  the  voltage  coil  of  the 
wattmeter  be  of  entirely  negligible  inductance  and  capacity,  or, 
where  the  voltage  and  current  of  the  circuit  are  harmonic,  that 
these  constants  be  so  mutually  adjusted  that  the  time  constant 
is  practically  zero.  If  this  is  not  the  case,  the  current  in  the 

voltage  coil  is  equal  to  — instead  of  where  E is  the 

i?i  Rx 

voltage  in  the  circuit,  0X  the  angle  of  lag  in  the  voltage  coil 

X 

which  is  dependent  on  the  relation  tan  6X  = aud  Rx  and  Xx 

At-i 

are  respectively  the  resistance  and  reactance  of  the  voltage 
coil.  The  currents  in  voltage  and  current  coils  now  have 


POWER,  POWER  FACTOR 


347 


a difference  of  phase  which  is  equal  to  0 — 0X  instead  of  #,  where 
0 is  the  angle  of  lag  in  the  main  circuit.  This  is  illustrated  by 
Fig.  210,  where  the  vectors  E and  I are 
the  impressed  voltage  and  the  current 
of  the  circuit  to  which  the  wattmeter  is 
attached,  and  Ix  is  the  current  flowing 
in  the  voltage  coil  and  lagging  0X  de- 
grees behind  the  voltage.  In  case  the 
voltage  coil  is  affected  by  capacity 
reactance,  the  angle  0X  is  negative, 
and  should  be  so  used  in  the  com- 
putations ; that  is,  current  Ix  of  Fig. 

210  would  lead  voltage  E and  the 
angle  between  Ix  and  I would  be- 
come 0 — ( — 0X)  = 6 + 0y  The  read- 
ing of  a wattmeter  in  which  the 
voltage  coil  is  reactive  is  therefore  proportional  to 

0X  cos  (0  — 0X),  R 


Fig.  210.  — Phase  Diagram 
showing  Relation  of  Currents 
and  Voltage  in  Wattmeter 
with  Reactive  Voltage  Coil. 
Lagging  Current  in  Main 
Circuit. 


IE  cos 


since 


T _ E cos  0A 
1\  ~ 


instead  of  — ■ 


Zx  = „ 

COS  <7X 


and  therefore 


A correct  reading  is  propor- 


Rx 

tional  to  IE  cos  6.  The  readings  of  such  a wattmeter  must 
therefore  be  multiplied  by  a factor  equal  to 

-p  _ cos  0 cos  0 

cos  0X  cos  {0  — 0j)  cos  0X  (cos  0 cos  0X  + sin  0 sin  dx)’ 

in  order  that  they  may  give  the  true  power.  This  multiplier 
may  be  called  the  “ correcting  factor  ” of  a reactive  wattmeter, 
and  is  given  here  with  the  following  discussion  for  the  purpose 
of  illustrating  the  relation  of  the  currents  in  the  coils  rather 
than  for  use  in  measurement. 


Since 


cos  0 - 


R 


VA2  + X2 


COS  0X  - 


R^ 


Vi^2  + X* 
the  correcting  factor  reduces  to 
RRX2  + Xx2R 


sin  0 = 


sin  0X  — 


X 


Vi?2  + X2' 

X 


Vi?x2  + X* 


1 + tan2  0X 
RRX  + XXxRx  1 + tan  0 tan  0X 


348 


ALTERNATING  CURRENTS 


The  formulas  show  that  when  tan 2 61  is  negligibly  small  (in 
which  case  9X  is  practically  equal  to  zero),  or  tan  91  is  equal  to 
tan  9 (in  which  case  9X  = 9 ),  the  correcting  factor  reduces  to 
unity,  and  the  readings  of  the  wattmeter  are  directly  proportional 
to  power.  When  tan  91  is  algebraically  smaller  than  tan  9,  the 
correcting  factor  is  less  than  unity,  and  the  wattmeter  reads  too 
high , and  when  tan  91  is  algebraically  greater  than  tan  9 , the  cor- 
recting factor  is  greater  than  unity,  and  the  wattmeter  reads  too 
low.  The  indications  of  a wattmeter  with  reactive  voltage  coil 
may,  therefore,  be  either  correct,  too  high,  or  too  low,  depend- 
ing upon  the  algebraic  value  of  the  time  constant  of  the  circuit 
upon  which  measurements  are  being  made.  When  9 = 90°,  the 
correcting  factor  is  equal  to  zero  and  the  true  power  is  zero 
even  though  the  wattmeter  gives  a finite  reading,  as  it  will  do 
under  these  circumstances  if  tan  91  has  any  appreciable  value, 
though  it  may  be  very  small.  As  a general  rule,  the  tan  9 
of  the  circuit  is  likely  to  be  positive 
and  greater  than  that  of  the  wattmeter 
coil,  so  that  the  readings  of  such  a 
wattmeter  are  generally  found  in  prac- 
tice to  be  too  high  ; but  in  ordinary 
measurements  it  is  impracticable  to 
make  an  advance  determination  of  the 
value  of  tan  9 , so  that  the  correcting 
factor  of  the  wattmeter  is  unknown.  If 


Fig.  211.  — Phase  Diagram  the  main  circuit  has  a leading  current, 
showing  Relations  of  Cur-  • •,  • •.  i .,  , 

. . w — is  a capacity  circuit, — and  the  vol- 

rents  and  Voltage  in  Watt-  1 J 

meter  with  Reactive  Vol-  tage  coil  of  the  wattmeter  is  reactive,  the 
tage  Coil.  Leading  Current  vector  relations  are  as  in  Fig.  211.  If 

in  Main  Circuit. 

the  voltage  coil  were  also  artected  by 
capacity,  9t  would  be  negative  and  71  would  lie,  in  Fig.  211, 
between  Jd  and  I. 


It  is  evident  that  the  correcting  factors  for  either  leading  or 
lagging  currents  depart  far  from  unity  when  9 approaches  90° 
and  that  the  indications  of  the  instrument  are  likely^  to  be  far 
from  accurate  under  those  circumstances  for  even  microscopic 
values  of  9V  Indeed,  as  a rule  ordinary  electrodynamometer 
wattmeters  are  not  to  be  depended  upon  for  accurate  measure- 
ment of  power  in  circuits  in  which  the  lead  or  lag  of  the  main 
current  approaches  90°. 


POWER,  POWER  FACTOR 


349 


The  only  safety  in  wattmeter  measurements  of  power  in  alternat- 
ing-current circuits , therefore,  lies  in  the  use  of  a wattmeter  with 
such  a very  small  time  constant  in  the  voltage  coil  that  it  may  he 
considered  absolutely  negligible.  This  construction  can  be  ac- 
complished by  placing  a relatively  very  large  non-reactive  re- 
sistance in  series  with  the  voltage  coil,  due  care  being  taken  to 
avoid  appreciable  capacity  between  conductors  of  the  resistance 
coil,  and  also  by  making  proper  use  of  the  mutual  induction 
between  the  coils.*  It  is  very  seldom  that  there  is  appreciable 
self -inductance  in  the  current  coil  of  a wattmeter;  but  if  there 
is,  the  effect  merely  increases  the  current  lag  in  the  main  circuit 
or  decreases  its  lead  by  a small  amount  and  is  therefore  taken 
into  account  in  the  above  discussion.  In  ordinary  commercial 
instruments  the  capacity  of  the  current  coil  is  insignificant. 

If  the  voltage  is  not  sinusoidal,  the  inductance  of  the  watt- 
meter will  cause  a reactance  having  a value  for  each  harmonic 
proportional  to  the  frequency  of  the  harmonic.  Thus  the  har- 
monics of  current  through  the  voltage  coil  will  each  have  dif- 
ferent phase  angles  when  compared  with  the  harmonics  of  the 
impressed  voltage.  As  the  power  is  equal  to  the  sum  of  the 
products  of  the  effective  values  of  the  current  and  voltage  of 
each  harmonic  with  the  cosine  of  their  phase  difference,  or 
^EnIn  cos  9n , it  is  evident  that  a separate  correcting  factor  is 
necessary  for  each  harmonic.  The  large  effect  of  any  reactance 
on  tire  higher  harmonics  may  cause  extremely  untrue  indica- 
tions of  a wattmeter  with  reactive  voltage  coil  where  it  is  used 
to  measure  power  in  a circuit  of  low  power  factor.  This  may 
occur  with  values  of  the  voltage  coil  reactance  that  might  be 
considered  negligible  when  the  voltage  and  current  are  sinu- 
soidal. The  relation  cos  6 = F cos  61  cos  (d  — 9f)  does  not  hold 
if  equivalent  sinusoids  are  substituted  for  the  irregular  currents 
in  the  current  and  voltage  coils  and  for  the  line  voltage,  for  it 
will  be  found  that  (0  — Of)  + 01  > 9.  Under  these  circum- 
stances the  current  vector  f of  Fig.  210  cannot  be  in  the  plane 
of  the  vectors  I and  E,  but  the  three  vectors  must  form  a tri- 
angular pyramid  in  which  the  faces  give  the  angles  IOE  = 9, 
IOI1  — 9—9v  and  I1OE=9v  A like  relation  holds  for 
Fig.  211. 

b.  Correction  of  Wattmeter  Readings  on  Account  of  the  Poiver 

* Art.  23. 


350 


ALTERNATING  CURRENTS 


Fig.  212.  — Watt- 
meter connected 
so  that  the 
Power  used  in 
Voltage  Coil  is 
added  to  that  of 
the  Circuit  being 
Tested. 


Absorbed  by  the  Instrument.  Another  correction  due  to  the 
power  used  by  the  wattmeter  itself  is  also  necessary.  Thus,  if 
the  voltage  coil  is  connected  to  the  circuit  be- 
tween the  current  coil  and  the  test  circuit  (Fig. 
212),  it  is  evident  that  the  power  measured 
includes  that  absorbed  by  the  voltage  coil.  If 
the  current  coil  is  included  between  the  point 
of  connection  of  the  voltage  coil  and  the  test 
circuit  (Fig.  213),  the  power  measured  includes 
that  absorbed  by  the  current  coil.  In  either 
case  this  power  should  be  small  and  usually 
may  be  neglected  ; but  when  this  is  not  the 
case,  it  is  easily  determined  from  the  resistance 
of  the  coil  included,  if  the  voltage  or  current  is 
known.  In  some  wattmeters  a special  correct- 
ing coil  wound  over  the  series  coil  is  introduced 
in  series  with  the  voltage  coil,  which  corrects 
for  the  current  in  the  voltage  coil, — the  in- 
strument being  connected  as  in  Fig.  212.  (Example  : Weston 
wattmeter.) 

c.  Effect  of  Eddy  Currents  in  Wattmeter 
Frame.  As  in  the  case  of  any  electrodyna- 
mometer or  other  instrument  operated  by 
electrodynamic  action,  it  is  necessary  that  a 
wattmeter  of  the  type  here  discussed  shall 
have  no  metal  in  its  frame  in  which  eddy 

currents  may  be  developed.*  If  this  precau-  . 

tion  is  not  carefully  looked  after,  the  constant 
of  the  instrument  will  vary  with  the  fre- 
quency, and  a calibration  is  necessary  for  every 
frequency. 

The  magnetic  effects  of  any  eddy  currents 
which  are  set  up  in  the  metal  supports  of  an 
instrument  may  have  quite  a marked  effect 
upon  the  magnetic  fields  around  the  coils,  and  as  the  intensi- 
ties of  the  eddy  currents  are  dependent  upon  the  frequency 
of  the  inducing  current,  the  above  statement  about  calibration 
is  evidently  correct.  For  a properly  built  wattmeter,  as  said 
above,  which  is  used  at  a point  near  which  there  are  no  masses 

* Art.  111. 


Fig.  213.  — Watt- 
meter connected 
so  that  the  Power 
used  in  the  Cur- 
rent Coil  is  added 
to  that  of  Circuit 
being  Tested. 


POWER,  POWER  FACTOR 


351 


of  metal,  a single  calibration  with  direct  currents  is  suffi- 
cient. 

2.  Electrometer  Method.  If  the  two  pairs  of  quadrants  of  a 
quadrant  electrometer  are  connected  with  points  of  potential, 
respectively,  Vx  and  V2,  and  the  needle  is  connected  with  a 
point  of  potential  V3,  then  the  deflection  of  the  needle  is  theo- 
retically 

ri  + r? 


Fig.  214.  — -Diagram  of  Electrom- 
eter Connections. 


d = k (T\  - r2)  [ys  - 

where  k is  the  constant  of  the  electrometer.  An  explanation 
of  the  truth  of  this  formula  may  be  made  as  follows.  In  Fig. 
214  the  electrometer  needle  is  well  under  the  quadrants,  and,  if 
the  needle  mov.es  slightly,  the  capac- 
ity afforded  by  the  quadrants  toward 
which  the  needle  moves  and  the 
portion  of  the  needle  under  them  is 
increased  slightly,  due  to  the  greater 
surface  exposed  to  the  inductive 
effect,  while  the  quadrants  from 
which  the  needle  moves  and  the  por- 
tion of  the  needle  influenced  by  them 
decrease  in  capacity.  If  c is  the 
change  in  capacity  of  either  pair  of 
quadrants  and  the  accompanying 
portion  of  the  needle  for  each  very  small  angular  movement  of 
the  needle,  then  the  amount  of  charge  gained  by  the  quadrants 
toward  which  the  needle  moves  is  ca  (Es  — V1')  and  by  the 
needle  is  ca  (F[  — V3~),  where  a is  the  angular  distance  moved. 
Likewise,  the  charge  lost  by  the  quadrants  from  which  the 
needle  moves  is  ca  ( Trs  — V^),  and  that  lost  by  the  needle  is 
ca(V2—V3).  These  expressions  come  from  the  fundamental 
equation,  Q=  CE.  Equal  charges  of  opposite  signs  must  flow 
to  quadrants  and  needle. 

It  has  been  proved  earlier  that  the  work  stored  in  a charge 
is  Q V,  where  V is  the  absolute  potential  of  the  charge. 
Therefore,  the  energy  gained  by  the  first  pair  of  quadrants  is 
I ca  (Fg  — Vx ) Vv  and  by  the  needle  ca  (Vx  — Tr3)  V3.  Like- 
wise, the  energy  lost  by  the  second  pair  of  quadrants  is 
\ ca  (V3  — V2 ) V2,  and  by  the  needle  \ ca  ( V2  — V3)  V3.  Sub- 
tracting the  energy  lost  from  the  energy  gained,  there  results 


352 


ALTERNATING  CURRENTS 


OT  = c«(F1-F2)(r3-SAiS). 

but  the  couple  tending  to  turn  the  needle  is  equal  to  the  work 
done  divided  by  the  angular  displacement.  Therefore,  the 
torque  _F,  tending  to  turn  the  needle,  is 

T')(r8-G±iSj. 

If  the  angle  « is  small,  so  that  the  surface  distribution  of  the 
charges  is  not  materially  altered,  this  torque  should  be  constant 
throughout  the  range  of  the  deflection  ; and  when  the  restrain- 
ing force  of  the  needle  suspension  is  proportional  to  the  angle 
of  deflection,  the  deflection  is  proportional  to  the  applied  torque. 
As  a result, 

d = k{v1-v2)(r3-^±Hy 

as  stated  above.  These  conditions,  however,  cannot  be  rigor- 
ously obtained  in  ordinary  electrometers  because  the  distribution 
of  the  charges  will  not  remain  unchanged  during  the  operations. 

If  vv  vv  and  v3  represent  the  instantaneous  values  of  the 
potentials  at  the  points  when  varying  alternatingly,  similar 
reasoning  shows  that  the  deflection  becomes 

d f Oi  - ”2')  (v3  - dt- 

If  it  is  desired  to  measure  the  power  absorbed  by  a reactive 

circuit,  the  electrometer  may  be 
used  in  the  following  manner : 
The  reactive  resistance  BC  is 
connected  in  series  with  the  non- 
reactive resistance  AB  (Figs. 
215  and  216).  Let  the  potential 
of  the  points  A , B , and  C at  any 
instant  be  represented  respec- 
tively by  vv  vv  and  v3 , when  B is 

Fig.  215.— Electrometer  in  First  Posi-  the  junction  between  the  react- 
tion  for  obtaining;  a Power  Reading.  . , , . . . 

ive  and  non-reactrve  resistance. 

Then,  if  a quadrant  electrometer  is  connected  with  its  quad- 
rants to  A and  B , and  its  needle  and  case  to  C (Fig.  215),  the 
deflection  is 


POWER,  POWER  FACTOR 


353 


d = ^rSo 

If  the  connection  of  the  needle  is  interchanged  so  that  it  is 
connected  to  B while  the  connections  of  the  quadrants  remain 
unchanged  (Fig.  216),  this  becomes 

d' = -f  X^’1  ~ ~ 

By  subtraction,  this  results  in 

k CT 

d'  -d=  — Jo  Oq  - v2)(v2  - v^dt. 


Dividing  this  by  kR , where  R is  the  resistance  of  AB , gives 

d'  — d _ 1 rTvx 
tX  r 


kR 


^(*’2 


v^)dt. 


Now  12  is  equal  to  the  instantaneous  value  of  the  cur- 
R 

rent  passing  through  the  circuit,  and  v%  — v3  is  the  correspond- 
ing instantaneous  value  of  the  voltage  between  the  terminals 
of  the  reactive  resistance  BO. 

d'  — d l CT  • 

Consequently,  ^ = — J iedt  = P,  where  P is  the  power 

absorbed  by  the  reactive  part  of  A B c 

the  circuit.  AAAAAAn — 

On  account  of  structural  de- 
fects, the  deflections  of  electrom- 
eter needles  do  not  always 
follow  the  theoretical  law,  as 
aforesaid.  Consequently,  it  is 
necessary  to  determine  how  great 
the  deviation  is  before  the  instru- 
ment may  be  relied  upon.  Or,  Fig.  216.  — Electrometer  ill  Second 
the  instrument  may  be  calibrated  Posi*!on  for  obtaining  a Power 
by  the  use  of  direct  currents 

passing  through  known  resistances,  which  are  so  adjusted  that 
vv  vv  and  v3  are  nearly  the  effective  values  of  the  tests. 

3.  Electrostatic  Wattmeter.  A modification  of  the  quadrant 
electrometer  may  be  made  which  reads  directly  as  a watt- 


354 


ALTERNATING  CURRENTS 


meter.*  In  this  case  the  needle  box  is  divided  diametrically 
into  two  parts  instead  of  into  quadrants.  The  needle  consists 

of  a disc  divided  diametrically 
into  two  parts  (Fig.  217).  The 
points  A and  B of  the  circuit  are 
connected  to  the  two  halves  of 
the  needle,  and  B and  C to  the 
two  halves  of  the  needle  box. 

Then  the  force  which  causes 
the  deflection  of  the  needle  is 
theoretically  proportional  to 
the  difference  which  is  found  by 
subtracting  the  total  loss  of 
energies  of  the  quadrants  and  needles  from  the  total  gain  due 
to  a small  movement,  exactly  as  was  done  in  the  previous  case 
for  the  electrometer. 

Hence, 

75  d 1 Br(v  1—  fo)/-  n. 


Fig.  217. — Diagram  of  an  Electro- 
static Wattmeter. 


since 


--1-— — : ^ is  the  instantaneous  current  and  (p2  — 1>3)  the 


R 


instantaneous  voltage  in  the  circuit. 

This  instrument  may  also  be  calibrated,  as  explained  above, 
by  passing  a known  direct  current  through  a known  resistance. 
Wattmeters  of  this  type 
have  been  designed  and 
constructed,  but  they  are 
not  very  satisfactory. 

4.  Three-voltmeter  Meth- 
od.f  As  in  the  previous 
method,  a non-reactive  re- 
sistance must  be  connected 
in  series  with  the  reactive 
circuit  to  be  tested  (Fig. 

218).  Non-reactive  voltmeters  are  then  respectively  connected 
between  the  points  A and  B , B and  (7,  and  A and  C.  Letting 


'r ~©~ 

r— ^ 

I 

k — (^y — 1 — (a^) — < 

Fig.  218.  — Connections  for  Three-voltmeter 
Measurement  of  Power. 


* Gerard’s  Lemons  sur  V Blectricite,  3d  ed.,  vol.  I,  p.  611  ; Hospitalier’s 
Traite  de  V iSnergie  Blectrique,  vol.  I,  pp.  205  and  507. 

t Suggested  by  Ayrton  and  Sumpner,  London  Electrician,  vol.  20,  p.  736; 
Electrical  World , vol.  17,  p.  329. 


POWER,  POWER  FACTOR 


355 


ev  e2,  and  e represent  instantaneous  voltages  at  the  three  volt- 
meters, then  e = e1  + ev  whence 


p*  p & p & — V p p 

O C-i  on  — 01  Z', 


1D2* 


But  the  instantaneous  value  of  the  power  in  the  inductive 

• £ 

circuit  is  p = ie2  = e2.  Substituting  the  value  of  e1e2  already 

R ' 

found  gives  p = — (e2  — <q2  — e22),  and  the  mean  power  ab- 
2 R 

sorbed  during  a period  is 

p = = dUC<^  ~ e'2  - ^ 


where  E,  Ev  and  E2  are  the  respective  readings  of  the  volt- 
meters. If  R is  not  known  and  the  value  of  the  current  I is 
known,  the  formula  may  be  written 

The  letters  E representing  the  effective  voltages  replace  the 
letters  e representing  the  instantaneous  voltages  in  the  formula 
after  the  integration  is  performed  on  the  formula,  because 


— f e2dt  = av  • e2=  E2, 

tJo 


as  has  already  been  proved.* 

In  order  that  the  results  of  measurements  by  this  method 
may  be  the  most  accurate  possible,  Ex  should  equal  E2,  which 
makes  the  method  inconvenient  for  use  in  ordinary  testing. 
Neither  is  the  method  sufficiently  accurate  to  compensate  for 
its  disadvantages.  The  accuracy  of  any  particular  measure- 
ment made  by  this  method  may  be  checked  by  inserting  a 
known  non-reactive  resistance  in  place  of  the  reactive  cir- 
cuit. This  method,  and  methods  5 to  8,  inclusive,  are  not 
desirable  for  ordinary  measurements  and  only  prove  useful 
under  special  conditions. 


* Art.  5. 


356 


ALTERNATING  CURRENTS 


5.  Three-ammeter  Method.  Instead  of  putting  a non-re- 
active resistance  in  series  with  the  reactive  circuit,  it  may 

be  placed  in  parallel  with 
it.  In  this  case  ampere- 
meters must  replace  the 
voltmeters  of  the  preced- 
ing method  (Fig.  219). 
One  amperemeter  meas- 
ures the  whole  current  /, 
another  measures  the  cur- 
rent Jj  in  the  non-reactive  resistance  B,  and  another  meas- 
ures the  current  I2  in  the  reactive  circuit.  It  is  evidently 
essential  that  the  amperemeters  are  of  negligible  reactance. 
Supposing  i,  iv  i2  are  the  instantaneous  values  of  the  currents 
at  any  moment,  we  have 


Pig.  219.  — Connections  for  Three-ammeter 
Measurement  of  Power. 


i = il  + i2  and  i2  — i2  — i2  = 2 ipv 
while  p = ei2  — Rip2  = ^ (i2  — — t22). 

Whence  P = ^ pdt  = ^{I2-I2- I2^>. 


This  may  also  be  written 

-1  Ji 

In  this  case  the  greatest  accuracy  is  given  when  and  I2  are 
about  equal;  but,  at  the  best,  the  method  is  not  very  exact. 
Its  accuracy  may  be  checked,  as  in  the  previous  case,  by  replac- 
ing the  reactive  circuit  by  a 
suitable  non-reactive  resist- 
ance. 

6.  Other  Three-instrument 
Methods.  Various  modifica- 
tions of  the  last  two  methods 
have  been  suggested  by  Ayr-  pIG  ooo.  — Connections  for  measuring 
ton,  Sumpner,  Blakesley,  and  Power  using  Two  Ammeters  and  One 

others.  One  of  the  obvious  Aoltnuter- 

arrangements  is  to  omit  the  amperemeter  in  series  with  the 
non-reactive  resistance  of  the  fifth  method,  and  connect  a volt- 


POWER,  POWER  FACTOR 


357 


meter  across  the  circuit  as  in  Fig.  220.  In  this  case  the  power 


becomes 


7.  Split  Dynamometer  Methods.  If  separate  alternating 
currents  of  the  same  frequency  are  passed  through  the  two  coils 
of  an  electrodynamometer,  its  reading  will  be  proportional  to 

1 CT  • 

— J ip2dt.  This  is  equal  to  IXI2  cos  6.  For 

= V2  /j  sin  a and  i2  = V2  I2  sin  (a  — 0 ), 

where  0 is  the  angle  between  the  two  current  waves, 
stituting  gives 

\ rr  i)  r*T 

•y  Jo  iihdt=JiJ  J2  sin  a sin  (a  - 0)  dt, 
which  is  equal  to 

- § sin  a sin  (a  — 0 ) da  = IXI2  cos  0. 


Sub- 


An  electrodynamometer  used  in  this  manner  is  called  a Split 
dynamometer.  Now  suppose  we  determine  the  values  of  Ix  and 
I2  by  means  of  a dynamometer  used  as  an  amperemeter  or  by 
other  instruments,  then  the  value  of  cos  0 is  at  once  found.  If 
the  measurements  are  all  made  by  the  same  electrodynamometer, 
its  constant  does  not  need  to  be  known.  Suppose  the  readings 
in  the  two  circuits  are  Jx2  = Jc$v  and  /22  = &S2,  and  the  reading  as 


a split  dynamometer  is  IXI2  cos  0 = 7cS3,  then 
This  plan  was  first  suggested  by  Blakesley.* 

8.  Blakesley  s Split  Dyna- 
mometer Method.  Blakesley 
planned  various  methods  for 
using  a split  dynamometer  in 
measuring  the  power  absorbed 
by  a reactive  circuit,  f In 
one  of  the  methods  a non- 
reactive resistance 


cos  0 - — -3 — . 

vs,s2 


Fig. 


221.  — Measurement  of  Power  by  a 
Split  Dynamometer. 

is  con- 
nected in  parallel  with  the  reactive  circuit  to  be  tested  (Fig. 
221),  and  a split  dynamometer  is  connected  so  that  one  coil 


* Alternating  Currents  of  Electricity,  2d  ed.,  p.  97. 
t Phil.  Mag.,  vol.  31,  p.  346. 


358 


ALTERNATING  CURRENTS 


carries  the  total  current,  and  the  other  carries  the  current  of 
the  reactive  branch.  An  amperemeter  is  also  placed  in  the 
reactive  branch.  Calling  i the  instantaneous  value  of  the 
total  current,  and  iv  i2,  respectively,  the  instantaneous  currents 
in  the  non-reactive  and  reactive  circuits,  the  following  re- 
lations hold:  the  reading  of  the  split  dynamometer  is  propor- 
tional to  1 I2  cos  9,  and  that  of  the  amperemeter  gives  I2\  but 

i\h  = 0 h)h  = ~ *22’  anc^  Rhh  = — f22)-  edua^ 

to  the  instantaneous  voltage  between  the  terminals  of  the  non- 
reactive resistance,  and  therefore  Ri1i2  is  equal  to  the  instan- 
taneous value  of  the  power  absorbed  by  the  reactive  circuit. 
Integrating  gives 


= R(1 I2  cos  9)  - Rif  = kRD  - Rif, 


where  D is  the  scale  reading  of  the  split  dynamometer,  and  k is 
its  constant.  Hence,  the  power  absorbed  by  the  reactive  cir- 
cuit is  equal  to  R times  the  difference  between  the  reduced 
split  dynamometer  reading  and  the  square  of  the  current  in  the 
reactive  circuit. 

A similar  result  may  be  gained  by  putting  the  amperemeter 
in  the  non-reactive  branch,  provided  the  instrument  is  itself 
non-reactive. 


CHAPTER  VIII 


POLYPHASE  CIRCUITS  AND  THE  MEASUREMENT  OF 
POWER  THEREIN 


100.  Polyphase  Systems.  — A Polyphase  system  is  a system 
comprising  a number  of  simple  alternating-current  circuits 
carrying  currents  of  different  phases  used  conjointly  to  obtain 
the  advantage  of  combined  vector  relations.  When  the  simple 
circuits  number  more  than  two,  the  several  line  voltages  are 
capable  of  forming  a closed  vector  polygon,  and  the  several 
line  currents  of  likewise  forming  a closed  vector  polygon.  The 
circuits  of  such  a system  may  be  operated  as  separate  single- 
phase circuits,  or  jointly  as  a polyphase  circuit. 

A polyphase  system  is  said  to  be  Balanced  when  the  voltages 
of  the  principal  single-phase  circuits  are  numerically  equal  to 
each  other  and  alike  in  form,  and  the  currents  of  the  same  cir- 
cuits are  also  numerically  equal  to  each  other  and  alike  in  form; 
with  the  additional  proviso  that  when  the  simple  circuits  number 
only  two,  the  difference  in  phase  between  the  two  voltages  is 
90°  and  the  difference  in  phase  between  the  currents  is  also  90°. 
When  the  number  of  single  circuits  composing  the  balanced  poly- 
phase system  is  greater  than  two,  the  phase  difference  between  the 

2 7 r 

successive  voltages  or  successive  currents  is  — , where  n is  the 

n 

number  of  phases  ( i.e . the  number  of  simple  circuits  composing 
the  polyphase  system).  Polyphase  systems  are  usually  operated 
with  either  two  currents  with  approximately  90°  difference  of 
phase,  called  quarter-phase  or  two-phase  currents,  or  three  cur- 
rents with  approximately  120°  phase  difference,  called  three- 
phase  currents. 

The  transmission  circuits  for  quarter-phase  currents  may  be 
arranged  to  be  entirely  independent  of  each  other,  four  wires 
being  then  required  (Fig.  37)  ; or,  three  wires  may  be  used,  in 
which  case  one  of  them  is  common  to  the  two  currents  (Fig.  38) 
and  the  current  in  the  third  or  common  wire,  at  any  instant,  is 
equal  to  the  algebraic  sum  of  the  currents  in  the  other  two. 

359 


360 


ALTERNATING  CURRENTS 


The  algebraic  sum  of  the  instantaneous  currents  in  the  three  wires 
is  always  equal  to  zero , the  total  return  current  being  of  course 

equal  to  the  total 
outgoing  current 
at  every  instant. 
The  effective 
current  in  the 
common  return 
wire  is  equal  to 
the  vector  sum  of 
the  two  circuit 
currents ; and  it 
is,  therefore, 
a/2  Z,  where  I is 
the  effective  cur- 
rent in  one  cir- 
cuit, provided  the  currents  are  equal  in  the  two  circuits  and 
have  a phase  difference  of  90°,  which  is  the  condition  when  the 
system  is  properly  designed  and  sym- 
metrically loaded  or  Balanced.  The 
voltage  between  the  two  outside  wires  of 
the  quarter-phase  system  with  common 
return  is  the  vector  sum  of  the  two  cir- 
cuit voltages,  and  is,  therefore,  a/2  E in 
a balanced  system,  where  E is  the  voltage 
between  one  side  and  the  common  return. 

The  common  current  in  a balanced  sys- 
tem is  45°  from  the  phase  of  the  current 
in  either  of  the  independent  wires  and 
the  joint  voltage  is  45°  from  the  voltage 
of  either  phase.  Figure  222  shows  the 
graphical  composition  of  the  voltages. 

A and  B are  the  two  line  voltages,  and 
R is  the  resultant  voltage  measured 
across  the  outside  wires. 

The  coils  of  quarter-phase  machines 
may  be  entirely  independent  of  each 
other,  in  which  case  an  armature  requires 
four  circuit  terminals,  or  the  circuits 
may  be  joined  so  as  to  require  only  three  terminals.  In  a 


Fig.  223.  — Methods  of  Con- 
nection— Two-phase  Arma- 
tures. 


Fig.  222.  — Voltage  Curves  of  Three-wire  Two-phase  System. 


POLYPHASE  CIRCUITS  AND  MEASUREMENT  POWER  3G1 


quarter-phase  machine  with  rotating  armature  the  armature 
may  be  wound  with  the  equivalent  of  a series-path  direct- 
current  winding,  and  four  collector  rings  and  independent 
circuits  are  then  required  to  avoid  short-circuiting  portions  of 
the  armature.  Figure  223  shows  diagrammatically  various 
ways  of  connecting1  the  coils  of  quarter-phase  machines.  (See 
also  Art.  22.) 

It  is  possible,  in  three-phase  systems,  to  use  three  entirely 


Fig.  224.  — Three-phase  Delta  Connection. 

independent  circuits,  each  consisting  of  two  wires,  and  carrying 
currents  of  120°  difference  of  phase  ; but  in  practice  the  circuits 
are  almost  invariably  combined  so  as  to  use  three  line  wires. 

The  three  armature  circuits  of  three-phase  machines  may  be 
connected  together  so  that  they  form  the  sides  of  a delta  with 
the  transmission  wires  connected  to  the  three  corners  of  the 
triangle  (Fig.  224),  or  one  end  of  each  of  the  three  circuits 


Fig.  225.  — Three-phase  Wye  Connection. 

may  be  individually  connected  to  the  transmission  wires,  the 
free  ends  of  the  three  circuits  being  connected  together  (Fig. 
225).  In  either  case  the  number  of  transmission  wires  is  three. 


362 


ALTERNATING  CURRENTS 


Fig.  226.  — Vector  Relations  in  Star-connected  System. 


POLYPHASE  CIRCUITS  AND  MEASUREMENT  POWER  303 


and  the  algebraic  sum  of  their  instantaneous  currents  is  always 
equal  to  zero. 

In  the  latter,  called  the  Wye  or  Star  arrangement,  which  is 
represented  by  the  symbol  Y,  the  voltage  between  anjT  two  line 
wires  in  a balanced  system  is  V3  E,  where  E is  the  voltage  in 
one  circuit  of  the  machine.  Thus,  in  Fig.  226  a,  the  triangle 
ABC  represents  the  vector  polygon  of  line  voltages,  and  the 
arrowheads  the  directions  of  the  vectors  which  are  of  successively 
120°  difference  in  phase.*  The  phase  diagram  of  these  vector 
voltages  (denominated  E AC,  ECB , and  EBA)  is  laid  out  from  the 
center  of  rotation  0 , with  the  vectors  of  the  same  dimensions 
and  directions  as  the  sides  of  the  triangle  ABC.  The  wye 
armature,  or  load  branch,  voltages  in  a balanced  system  are  OA, 
OB , and  00  (denominated  E0A,  E0B,  and  Euc ).  This  must  be 
the  case  since  the  three  branches  are  equal  and  meet  at  a central 
or  neutral  point,  and  their  free  terminals  have  the  same  instan- 
taneous potentials  as  the  points  A,  B , and  C.  It  is  evident,  then, 
from  Fig.  226  a that  the  branch  voltages  — E0A  and  E0B  form  a 
closed  vector  triangle  with  line  voltage  EBA , and  that  the  vector 
difference  of  E0A  and  E0B  (i.e.  the  vector  sum  of  E0A  and 

— Eob)  is  therefore  equal  to  EBA  f;  likewise,  branch  voltages 

— Eob  and  Eoc  form  a closed  vector  triangle  with  line  voltage 
Ecb.  and  Ecb  is  therefore  equal  to  the  vector  sum  of  E0B  and 

— J£oc>  and,  in  the  same  manner,  it  will  be  observed  that  EAC  is 
equal  to  the  vector  sum  of  — E0A  and  Euc.  It  is  sometimes 
more  convenient  to  illustrate  these  relations  by  laying  down 

— Eob  as  at  OH , and  completing  the  parallelogram  on  the  sides 
Eoa  and  — Eob,  whence  it  is  at  once  seen  that  the  diagonal,  OB, 
is  equal  by  construction  to  EBA.  Consequently  EBA  must  be  the 
vector  sum  of  E0A  and  — E0B.  The  vector  — E0B  is  of  course 
the  same  as  vector  EB0.  The  line  OB  shows  the  position  of  EBA 
in  the  phase  diagram  of  branch  and  line  voltages  for  a balanced 
three-phase  Y-connected  circuit.  The  lines  OGr  and  OF7 likewise 
show  the  positions  of  EAC  and  ECB  in  the  phase  diagram. 

Figure  226  b shows  the  vector  polygon  of  the  voltages  E0A, 
Eob,  and  Eoc,  which,  being  numerically  equal  and  120°  apart  in 
phase  for  a balanced  system,  make  a closed  equilateral  triangle. 

Returning  to  Fig.  226  a,  since  the  angle  BOA  in  the  phase  dia- 
gram equals  angle  OAB  and  OB  and  OA  are  of  equal  length,  it  is 
* Art.  103.  f Art.  79. 


304 


ALTERNATING  CUR  1 1 ENTS 


obvious  that  OD  = V3  OA  and,  therefore,  that  EBA  — V3  E0 , 
= VSEB0  when  these  are  given  in  effective  values.  Similar 
relations  holding  for  the  other  two  line  voltages,  it  is  seen  that 
El  = v/3  E,  when  EL  is  the  effective  value  of  a line  voltage 
and  E is  the  effective  value  of  a branch  voltage.  The  figure 
shows  that  line  voltage  EBA  leads  branch  voltage  E0A  by  30° 
and  branch  voltage  EUB  by  150°  ( i.e . lags  behind  EB0  by  30°); 
line  voltage  EAG  leads  branch  voltage  Euc  by  30°  and  branch 

voltage  E0ibj  150°; 
and  line  voltage  Eclt 
leads  branch  voltage 
Eub  by  30°  and  Eoc 
by  150°.  ' 

In  Fig.  227,  the 
curve  R shows  the 
potential  difference 
between  A and  B. 

The  line  current  in 
any  star-connected 
circuit  must  always 
be  the  same  as  that 
passing  through  the 
branch  in  which  the 
line  terminates. 

In  the  Delta  or  Mesh  winding,  which  is  often  represented  by 
the  symbol  A,  the  voltage  between  wires  is  evidently  that  gen- 
erated by  one  coil,  and  the  current  in  the  line  wire  is  the  com- 
bination of  those  in  two  adjacent  coils,  or  V3  I in  a balanced 
system,  where  I is  the  current  in  a coil.  Thus,  in  Fig.  226  c, 
the  triangle  3INP  is  the  vector  polygon  of  the  line  currents 
from  the  delta  corners  A , B,  and  C , of  Fig.  224,  the  arrow- 
heads representing  the  vector  directions.*  The  phase  diagram 
of  these  vector  currents,  called  IAl,  IBL,  and  ICL,  respectively,  is 
laid  out  from  the  center  of  rotation  0 with  the  vectors  of  the 
same  directions  and  dimensions  as  those  forming  the  sides  of 
the  triangle  MNP.  The  branch  currents  between  BA,  AC, 
and  CB , Fig.  224,  called  IBA,  IAC , and  ICB,  respectively,  are 
numerically  equal  and  of  successively  120°  difference  in  phase. 
But  IBA  and  IAC  join  with  IAL  at  the  delta  corner  A and  the 

* Art.  103. 


Fig.  227.  — Voltage  Curves  of  a Three-phase  System. 


POLYPHASE  CIRCUITS  AND  MEASUREMENT  POWER  365 

three  must  therefore  form  a closed  vector  triangle,  since  the  alge- 
braic sum  of  their  instantaneous  values  must  always  be  equal  to 
zero  in  accordance  with  Kirchoffs  law  of  currents  at  junction 
points.  Similar  relations  exist  between  the  branch  and  line  cur- 
rents at  the  other  corners  of  the  delta.  To  fulfill  these  conditions 
OR , OS,  and  OT  may  represent  the  vector  currents  IBA,  ICB,  and 
IAC.*  Then,  from  the  parallelogram  OTRU , it  is  seen  that  IAL 
is  the  vector  difference  of  IBA  and  I AC  (i.e.  the  vector  sum  of  IBA 
and  — /4C)  ; parallelogram  ORSV  shows  IBL  is  equal  to  the 
vector  sum  of  — IBA  and  ICB ; and  parallelogram  OSTW  shows 
that  ICL  is  equal  to  the  vector  sum  of  — ICB  and  IAC.  These 
relations  are  in  accord  with  the  principle  enunciated  by  Kirch  off 
that  the  algebraic  sum  of  the  currents  meeting  in  a point  is  zero, 
when  those  flowing  toward  the  point  are  considered  of  one  sign 
and  those  flowing  away  from  the  point  of  the  opposite  sign.  At 
any  point  of  meeting  such  as  A,  Fig.  224,  the  sum  of  the  instan- 
taneous currents  must  always  be  zero,  and  the  geometric  sum  of 
their  vectors  must  therefore  be  zero.  But  we  have  taken  IBA 
to  be  from  B to  A and  IAC  and  ICB  must  therefore  be  taken 
in  the  directions,  respectively,  from  A to  C and  C to  B. 
Hence  IBA  has  a direction  toward  the  junction  A,  and  IAC  has 
a direction  away  from  A and  must  be  reversed  in  order  that  it 
may  add  to  IBA  to  give  IAL.  This  is  expressed  in  the  vector 
triangle  OUR , where  the  sum  of  IBA  and  — IAC  is  vectorially 
equal  to  IAL,  or  IAL  = IBA  + (—  IAC').  The  same  relations 
exist  for  the  other  corners.  From  Fig.  226  c it  is  seen  that 
Il  = v3I,  where  IL  is  the  effective  value  of  a line  current 
in  amperes  and  I the  effective  value  of  a branch  current. 
It  is  also  seen  that  IAL  lags  behind  IBA  by  30°  and  IAC  by  150°, 
IBL  lags  behind  ICB  by  30°  and  IBA  by  150°,  and  ICL  lags  behind 
IAC  by  30°  and  ICB  by  150°.  Figure  226  d shows  a vector 
polygon  of  the  branch  currents. 

Figure  228  shows  ways  in  which  the  coils  of  three-phase 
machines  may  be  connected  to  the  external  circuit.  The 
arrangements  are  either  of  the  wye  or  delta  connection.  The 
point  of  common  connection  0 in  the  wye  arrangement  is 
called  the  Neutral  point  of  the  winding.  A line  wire  may  or 
may  not  be  led  from  this  point,  depending  on  the  conditions  of 
use  of  the  current. 


* Art.  103. 


366 


ALTERNATING  CURRENTS 


In  the  case  of  a two-phase  system,  it  is  obvious  that  the 
algebraic  sum  of  the  instantaneous  currents  must  always  be  zero 
at  any  section  taken  across  the  line  wires  when  the  two  phases 
are  kept  separate  by  the  use  of  four  main  line  wires,  because  the 
instantaneous  incoming  current  is  equal  to  the  instantaneous  out- 
going current  in  each  single-phase  circuit.  When  three  wires 
are  used  for  such  a circuit,  the  joint  wire  is  a common  return 
for  the  other  two,  and  the  algebraic  sum  of  the  instantaneous 
currents  in  the  three  wires  is,  therefore,  clearly  equal  to  zero. 


Fig.  228.  — Methods  of  Connection  of  Three-phase  Machines  or  Loads. 


In  other  polyphase  systems,  the  vectors  of  the  several  line 
currents  are  capable  of  making  a closed  vector  polygon,  and  it 
therefore  follows  that  at  any  section  taken  across  the  line  wires, 

ij  sin  a + /2  sin  («  — /S2)  + •••  + In  sin  (a  — /3„)  = 0, 

\ilv  Iv  -In  are  the  effective  line  currents  and  /32  •••  are  the 
angular  differences  between  the  several  current  vectors.  This 
must  be  equally  true  for  each  harmonic  of  current.  But  each 
of  the  terms  on  the  left-hand  side  of  this  equation  multiplied 
by  V2  gives  the  instantaneous  value  of  the  current,  and  the 
algebraic  sum  of  the  instantaneous  currents  is  therefore  always 
equal  to  zero. 

This  is  plainly  to  be  seen  in  the  case  of  star-connected  wind- 
ings, when  it  is  remembered  that  the  electric  current  acts  like 
the  flow  of  an  incompressible  fluid  and  the  aggregate  current 
flowing  from  the  neutral  point  through  one  or  more  of  the 
windings  must  at  each  instant  be  equal  to  the  aggregate  current 
entering  the  neutral  point  at  that  instant  through  the  other 
winding's.  This  is  in  accordance  with  what  is  called  Ivirchoff  s 
law  of  current  flow  at  a junction  point.  In  case  of  the  use  of 


POLYPHASE  CIRCUITS  AND  MEASUREMENT  POWER  367 


a neutral  wire,  or  other  extra  wire  in  a polyphase  system,  the 
instantaneous  current  flowing  therein,  across  the  section  taken, 
must  be  included  in  the  algebraic  sum. 

When  the  system  is  balanced  and  has  n phases, 


*3  — ^ m ^ ^ ! 

in  = im  sin  (a  - 2^-n~p— 

. . . . . i . f 2 

Hence,  ix  + *2  + «3  + •••  + in  = im  j sin  a sin  la 

. f 4 7T  \ . f 0(  w — 1 )*7t'\  ) 

but  evidently,  sin  a + sin  q.  sin  ^ a — + •• 


+ sin  l^a  — 2 

and  therefore  i1  + i2  + i3  + •••  +in  = 0. 


(n  — l)7r^  _ 


;0, 


101.  Uniform  Power  in  Polyphase  Systems.  — In  general,  the 
power  transferred  in  a balanced  polyphase  circuit  is  uniform 
throughout  each  period,  and  the  torque  exerted  by  balanced 
polyphase  machinery  is  uniform.  This  is  different  from  the 
conditions  in  single-phase  circuits,  where  the  power  has  been 
shown  to  vary  between  a maximum  and  a minimum  during  every 
quarter  period.*  In  the  case  of  a single-phase  circuit  the 
power  at  any  instant  is  imem  sin  (a  — 0)  sin  a = imem  sin2  a cos  9 
— imem  sin  a cos  a sin  0,  which  varies  with  a.  In  a balanced  two- 
phase  circuit  the  instantaneous  power  is  imem  sin2  a cos  9 + 
imem  sin2  (a  — 90°)  cos  0 = imem  cos  0 (sin2  a + cos2a)  = imem  cos  0, 
which  is  constant.  In  the  same  way  the  power  in  a balanced 
three-phase  circuit  is,  when  im  and  em  represent  maximum  values 
of  branch  currents  and  voltages,  imem  cos  0 {sin2  « + sin2  (a  — 120°) 
+ sin2  (a  — 240°)  ( = § imem  cos  9 , which  is  constant ; and,  in  gen- 
eral, the  power  in  any  balanced  polyphase  circuit  in  which  the 


* Art.  92. 


368 


ALTERNATING  CURRENTS 


9 Jr 

phase  differences  are  equal  to  - — , where  n is  the  number  of 

n 

phases,  is,  when  im  and  em  are  maximum  values  of  branch  cur- 
rents and  voltages, 

imem  cos  6 | sin2  « -f-  sin2^«  — ^ j + sin2^«  — + ••• 


which  is  equal  to  cos  6 , and  is  constant,  since 

s.  n ] 


sin  2tt  + sin2 


n J 


-f-  sin 


. 2{  2(w— l)7r\  n 

-f  sin2  a x — = - . 


This  being  true  for  one  harmonic  it  is  true  for  all  harmonics 
where  the  currents  and  voltages  in  a balanced  circuit  are  not 
sinusoidal. 

The  uniformity  of  power  in  a balanced  polyphase  circuit  may 
also  be  directly  deduced  from  the  proposition  that  the  resultant 
of  n equal  harmonic  motions  acting  in  lines  having  successive 


angular  differences  of  is  a uniform  circular  motion  with  an 

n 

amplitude  equal  to  - times  the  amplitude  of  the  components. 

mi 


It  is  to  be  observed  that  uniform  transfer  of  energy  is  in- 
herent in  direct  current  and  balanced  polyphase  alternating- 
current  circuits,  but  that  this  attribute  is  not  possessed  by 
single-phase  circuits  nor  in  general  by  polyphase  circuits  which 
are  unbalanced. 

102.  Delta  and  Wye  Connections  for  more  than  Three  Phases. — 

With  more  than  two  phases  the  number  of  line  wires  may  be 
equal  to  the  number  of  phases,  as  illustrated  in  Figs.  228  and 
229  a to  7j,  which  show  corresponding  delta  and  star  arrange- 
ments of  operating  circuits  and  the  connection  of  line  wires 
thereto,  for  various  numbers  of  phases.  Turning  to  the  illus- 
trations in  Fig.  229,  a to  li  inclusive,  which  show  simple  mesh 
and  star  connections  for  five,  six,  seven,  and  eight  phases,  it  will 
be  observed  that  there  are  always  as  many  corners  in  the  mesh 
connection  as  there  are  operative  circuits  or  phases,  and  it  is 
obvious  that  an  equal  number  of  line  wires  individually  con- 
* Todhunter’s  Plane  Trigonometry , p.  243. 


POLYPHASE  CIRCUITS  AND  MEASUREMENT  POWER  369 


j-  k. 

Fig.  22<J.  — Connection  of  Line  Wires,  for  Various  Numbers  of  Phases. 


370 


ALTERNATING  CURRENTS 


nected  to  these  corners  as  shown  will  convey  all  of  the  currents 
of  the  operative  circuits.  Each  line  wire  will  convey  the  vec- 
tor difference  of  the  currents  in  two  of  the  mesh  circuits.  That 
is,  the  current  in  line  wire  B is  the  vector  difference  of  currents 
in  a and  b,  and  so  on.  Four  wires  may  thus  be  used  for  a four- 
phase  system,  five  wires  for  a five-phase  system,  etc. 

When  the  star  arrangement  is  used,  one  end  of  each  operative 
circuit  lias  a line  wire  connected  to  it  and  the  minimum  prac- 
ticable number  of  line  wires  is  again  equal  to  the  number  of 
operative  circuits.  The  free  ends  of  the  operative  circuits  are 
connected  with  each  other  at  the  neutral  point,  and  each 
operative  circuit  utilizes  all  the  other  circuits  and  lines  as  its 
return  lines.  In  this  case,  the  current  in  a line  wire  is  equal  to 
the  current  in  the  operative  circuit  to  Avhich  it  is  connected,  but 
the  voltage  between  any  two  line  wires  is  the  vector  difference 
of  the  voltages  of  the  corresponding  two  operative  circuits. 

An  additional  line  conductor  may  be  led  from  the  neutral 
point,  if  desired,  and  this  may  then  act  as  a partial  return  wire 
for  all  the  circuits  in  case  the  system  is  unbalanced. 

Figure  229  i,  j,  and  k show  three  combinations  of  mesh  and 
star  loads,  so  that  six,  eight,  and  twelve  phase  loads  may  be  fed 
from  three,  two  (using  four  wires),  and  three  phase  lines  re- 
spectively ; the  value  of  such  combinations  is  explained  later. 

103.  Relations  between  Currents  and  Voltages.  — In  both  the 
graphical  and  analytical  representation  of  polyphase  currents  or 
voltages,  it  is  particularly  desirable  to  adopt  a notation  which 
logically  and  conveniently  indicates  the  vector  relations  in  the 
circuit.  This  may  be  accomplished  by  following  the  methods 
described  in  Chapter  V and  exemplified  in  Art.  100.  A diagram 
showing  the  connection  of  circuits  (whether  mesh  or  star)  may 
be  drawn,  and  phase  diagrams  of  currents  and  voltages  then 
laid  down.  The  connection  diagram  being  carefully  lettered, 
each  vector  in  a phase  diagram  may  be  distinguished  by  the  two 
corresponding  terminal  letters  taken  from  the  diagram  of  con- 
nections, and  the  vectors  in  analytical  expressions  may  be 
identified  by  the  same  letters  used  as  subscripts.  Vector 
voltages  may  be  considered  as  falling  in  the  direction  from  the 
point  corresponding  to  the  first  subscript  toward  the  point 
corresponding  to  the  second.  For  instance,  in  a circuit  from 
A to  B , where  A is  taken  as  the  point  of  positive  or  higher 


POLYPHASE  CIRCUITS  AND  MEASUREMENT  POWER  371 


potential,  the  voltage  is  EAB , the  positive  direction  being  from 
A to  B.  Likewise  vector  currents  may  be  considered  as  flowing 
from  the  point  corresponding  to  the  first  subscript  to  the  point 
corresponding  to  the  second  subscript  when  caused  to  flow  by 
the  difference  of  potential  of  those  two  points.  For  instance, 
in  a circuit  having  the  voltage  EAB  the  current  would  be  under 
these  circumstances  IAB.  If,  however,  the  current  is  to  be 
represented  as  flowing  from  the  point  of  low  to  the  point  of 
high  potential,  under  the  influence  of  an  electromotive  force 
in  the  circuit  itself  or  other  cause,  it  may  be  considered  as 
flowing  from  the  point  corresponding  to  the  second  subscript 
to  the  point  corresponding  to  the  first  subscript ; and  in  such 
a case,  the  circuit  having  a voltage  EAB , the  current  would  be 
IBA ; if  the  subscripts  are  not  changed,  this  may  be  written 
— Iabi  1-ba  = -^-ab* 


To  illustrate  further,  Fig.  230  shows  a wye-connected  three- 
phase  generator 
armature  and  the 
corresponding  phase 
diagram  of  voltages. 

The  voltages  of  the 
coils  a , b , c,  are  re- 
spectively EA0,  E b0, 

Ec0 , and  successively 
differ  120°  in  phase. 

The  terminal  vol- 
tages become,  vec- 
torially : 

Eab  — E_a0  + Eqb  — Eao  + ( — Ego")  > 

EjiC  — Ego  + Eqc  = EgU  + ( — ^_CO  ) j 
Eca  — Eco  -T  Eoa  = Eco  + ( — FT 1(J ). 


Fig. 


230.  — Wye  Connected  Armature  and  Phase 
Diagram  of  Voltages. 


The  currents  in  the  coils  are  IAoi  IBoi  and  Ic0,  and  lag  behind 
their  respective  voltages  by  angles  of  lag  fixed  by  their  respec- 
tive circuits. 

The  following  are  the  relations  between  the  currents  and  the 
voltages  in  the  lines  and  coils  of  a balanced  three-phase  system 
developed  from  the  earlier  discussion  and  illustrated  in  Fig.  228  : 

1.  Star  Connection.  Ilme  = Jeoil;  EAB  =■■  EBC  = ECA  = V3  Ecoil. 
Line  voltage  EAB  is  the  vector  sum  of  coil  voltages  E A0  and 
Eob  (note  that  EAO  = — EOA)  and  is  30°  behind  the  phase  of  coil 


372 


ALTERNATING  CURRENTS 


voltage  Eub.  Line  voltage  EAC  is  the  vector  sum  of  coil  vol- 
tages Ea0  and  Eoc  and  is  30°  behind  the  phase  of  coil  voltage 
Eau.  Similar  relations  hold  for  the  other  two  corners.  For 
instance,  line  voltage  ECA  (which  is  equal  to  — EAC~)  is  the 
vector  sum  of  coil  voltages  Eco  and  EOA  and  is  30°  behind  the 
phase  of  EOA  and  30°  ahead  of  Eco ; and  ECB  (equal  to  — EBC~) 
is  the  vector  sum  of  Eco  and  E0B , and  is  30°  behind  the  phase 
of  Ec0  and  30°  ahead  of  the  phase  of  EOB. 

_ 2.  Delta  Connections.  EAB  = EBC  = ECA  = EcoiU  IUne  = Vs  Jcoil. 
Ila  is  the  vector  sum  of  branch  currents  IAC  and  IAB  (=  — IBA ) 
and  is  30°  ahead  of  IAC.  ILB  is  the  vector  sum  of  branch  cur- 
rents IBA  and  IBC  ( = — ICB)  and  is  30°  ahead  of  IBA.  is  the 
vector  sum  of  branch  currents  ICB  and  ICA  ( = — IAC)  and  is  30° 
ahead  of  ICB. 

Figures  226  a and  226  c show  the  relations  that  exist  if  non- 
'reactive  wye  and  delta  circuits  are  placed  on  the  same  line  wires. 
Thus,  the  line  voltage  between  each  pair  of  wires  is  in  phase 
with  the  delta  current  between  the  same  wires,  so  that  IBA , ICB, 
and  IAC  of  c are  parallel,  respectively,  to  EBA,  ECB , and  E AC 
of  a.  Also,  the  voltage  and  current  in  each  branch  of  the 
wye  are  in  the  same  phase,  so  that  E0A , E0B , and  Eoc  of  a are 
parallel,  respectively,  to  IAL , IBL , and  ICL  of  c.  Also,  the  wye 
branch  voltages  in  a balanced  three-phase  circuit  successively 
differ  in  phase  from  each  other  by  angles  of  120°,  and  they 
occupy  certain  specific  relations  to  the  delta  or  line  voltages, 
which  are  illustrated  in  Figs.  226  and  230.  It  will  be  observed 
from  Fig.  226  a that  EBA,  which  is  in  phase  with  IBA,  leads  EOA  by 
30°.  But  it  will  also  be  observed  from  Fig.  226  c that  IAL  lags 
30°  behind  IBA,  and  IAL  is  therefore  in  the  same  phase  as  EOA. 
That  is,  the  line  current  flowing  from  a corner  of  the  delta  has 
the  same  phase  as  the  voltage  measured  from  the  neutral  point 
of  the  circuit  to  the  same  corner,  when  the  delta  circuit  is  non- 
reactive. Now,  the  branch  voltage  EOA  of  the  wye  circuit  is 
identical  with  the  voltage  E0A  measured  from  the  neutral  point 
to  the  corner  A of  the  delta  circuit,  and  they  are  therefore  in 
the  same  phase.  Hence,  I0A  of  the  wye  circuit,  which  is  in 
phase  with  E0A , is  also  in  phase  with  IAL  of  the  delta  circuit. 
By  the  same  reasoning,  IOB  of  the  wye  circuit  is  in  phase  with 
IBL  of  the  delta  circuit,  and  Ioc  of  the  wye  circuit  is  in  phase 
with  ICL  of  the  delta  circuit.  These  relations  are  always  true 


POLYPHASE  CIRCUITS  AND  MEASUREMENT  POWER  373 


for  balanced  wye  and  delta  circuits  when  the  phase  of  each 
branch  current  coincides  with  the  phase  of  the  corresponding 
branch  voltage. 

If  the  delta  and  wye  branches  of  the  balanced  circuits,  on  the 
other  hand,  are  reactive,  but  all  of  the  same  power  factor,  the 
voltage  and  current  in  each  branch  differ  in  phase,  but  the  dis- 
placement of  the  line  current  from  the  phases  of  the  branch  cur- 
rents combining  at  a delta  corner  is  not  altered,  and  the  delta 
line  currents  ZUne  therefore  are  still  in  phase  with  the  correspond- 
ing wye-branch  currents  Icojl.  If  the  power  factor  of  the  delta 
branches  differs  from  the  power  factor  of  the  wye  branches, 
the  delta-line  currents  ijine  are  then  out  of  phase  with  the  wye- 
branch  currents  _Tcoil,  and  the  main-line  currents  feeding  the 
wye  and  delta  jointly  are  vector  resultants  of  the  delta  and  wye 
line  currents. 

If  the  circuits  of  utilization  in  a polyphase  system  are  not 
machines  (for  instance,  are  incandescent  lamps),  the  devices 
must  be  connected  exactly  as  would  be  the  coils  of  a machine  ; 
unless  transformers  intervene,  in  which  case  the  secondary 
circuits  may  be  independent ; but  the  load  should  be  uniformly 
distributed  to  keep  the  system  balanced. 

The  following  examples  deserve  careful  study,  as  they 
exhibit  the  phase  and  quantity  relations  of  currents  and  vol- 
tages in  polyphase  circuits  which  are  unbalanced.  The  voltages 
and  currents  are  assumed  to  be  sinusoidal  to  avoid  undue  com- 
plexity in  exhibiting  the  underlying  principles.  If  irregular 
curves  are  dealt  with,  a rigorous  solution  requires  that  they 
be  analyzed  and  their  sinusoidal  components  be  used  in  the 
solution ; but  in  many  instances  the  substitution  of  equivalent 
sinusoids  for  the  irregular  waves  is  sufficientl}7  accurate.  The 
first  two  examples  deal  with  three-phase  unbalanced  wye 
loads,  in  which  the  position  of  the  neutral  point  of  the  wye 
with  reference  to  the  three-line  voltages  must  be  determined.* 

Example  1.  — A three-phase  line  (. ALBLCL,  Fig.  a)  is  con- 
nected to  a non-reactive  wye  circuit  containing  resistance  of 
one  ohm  in  the  branch  AO.  two  ohms  in  BO.  and  three  ohms 
in  CO.  The  voltage  is  balanced,  being  100  volts  on  each  phase. 

(a)  What  are  the  values  of  the  current,  voltage,  and  angle 
of  lag  in  each  branch? 

* H.  P.  Wood.  Electrical  World,  vol.  55,  p.  1597. 


374 


ALTERNATING  CURRENTS 


(S)  What  are  the  current  values  in  the  line  wires  AAL , BBL, 
and  CCL  and  their  phase  relations  respectively  with  the  vol- 
c tages  Eab,  Ebc,  and  ECA  ? 


CO  M 

A 

Eca:1") 

L 

A Rao=1  A 

1 ebc 

=10V 

/ 

/Rbo=2 

t 

Eab=ioo 

1 

B, 

B 

Example  1. 

— Figure  a. 

(c)  What  is  the  total 
power  absorbed  ? 

Figure  a diagrammati- 
cally  represents  the  load. 

(a)  The  triangle  ABC 
(Fig.  5)  is  drawn  to  scale 
representing,  by  the 
length  and  positions  of  its  sides,  the  line  voltages  EAB,  EBC,  and 
Eca.  For  convenience,  the  side  AB  is  made  horizontal.  The 
point  O'  is  taken  at  any  convenient  position  in  the  plane ; and 
the  lines  O' A,  O'B , and 
O'  C are  drawn,  representing 
the  corresponding  voltages 
EaA , EaB,  and  E^c  from  O' 
to  A,  B,  and  C.  The  posi- 
tion of  the  point  O’  is 
assumed  for  convenience  in 
the  process  of  determining 
the  location  of  the  neutral 
point  0 of  the  circuit,  with 
reference  to  the  points  A, 

B , and  C.  Rectangular  axes 
are  erected  with  the  origin 
at  O'  and  the  axis  of  abscissas  parallel  to  AB. 

From  Fig.  b we  obtain  the  following  vector  equations,  in 
which  the  true  values  of  a and  b are  to  be  derived : 


B0-a  = a -ft. 

Bo  b = — (100  - «)  — jb. 

Eq. v = a — 50  + j (86. 6 — 5). 


As  the  circuit  is  non-reactive,  the  directions  of  the  currents, 
I(yA,  I ob'  lav,  may  be  represented  by  the  lines  0'AV  0'BV  and 
0'CV  coincident  with  the  lines  O' A,  O'B.  and  O' C.  Dividing 
each  vector  equation  for  voltage  given  above  by  the  resistance 
of  its  respective  branch,  gives : 


POLYPHASE  CIRCUITS  AND  MEASUREMENT  POWER  375 


Since  the  algebraic  sum  of  the  instantaneous  currents  meeting 
at  the  neutral  point  of  the  system  (currents  flowing  towards 

the  neutral  point  being  taken 
of  one  algebraic  sign  and  those 
flowing  away  being  taken  of 
the  other  sign)  must  always 
reduce  to  zero,  it  is  plain  that 
the  algebraic  sum  of  the  scalar 
values  of  the  three  vertical 
components,  and  likewise  of 
the  three  hori- 
zontal compo- 
nents, in  these 
three  vector  equa- 
tions must  each 
reduce  to  zero.  This  also  fol- 
lows from  the  fact  that  the 
three  currents  of  a three- 
phase  system  make  a closed 
vector  triangle,  and  therefore 
In,  + InR  + Ior  = 0.  To  cor- 

Ex ample  1.— Figure  c.  0A  B oc 

rectly  locate  the  neutral  point 
it  is  therefore  only  required  to  solve  for  a and  b in  the  following 
equations : 50 

*-hr-i  +i-T  = #> 


and 

whence 


V 2 2/  3 

, b 86.6  b n 
-4-5+~-3  = °; 

a =36.4  and  5 = 15.8. 


The  values  of  the  wye  currents  and  the  location  of  the  neutral 
point  0 are  determined  by  substituting  these  values  in  the  fore- 
going vector  equations  of  current,  and  are  plotted  in  Fig.  c. 


376 


ALTERNATING  CURRENTS 


In  this  figure  the  three  line-voltages  are  also  drawn  from  0 , 
making  a phase  diagram.  The  numerical  values  of  the  voltages 
E0A,  Eob,  and  Eoc  are  determined  by  substituting  the  values 
of  a and  b , just  found,  in  their  vector  equations  and  obtaining 
the  tensors  in  the  usual  manner,  thus  : 


Eoa  = V (36.4 ) 2 + (15. 8)2  = 39.7  volts. 
Eob  — ^ (63. 6)2  4-  (15. 8)2  = 65.5  volts. 


Eoc  = V(13.6)2  + (70. 8)2  = 72.1  volts. 

The  numerical  values  of  the  currents  are  found  in  the  same 
way  from  their  equations,  or  by  dividing  the  voltages  just 
determined  by  the  accompanying  branch  impedances  with  the 
result  that 

IOA  — 39.7  amperes. 

I0B  = 32.8  amperes. 

Ioc  = 24.0  amperes. 


The  angles  of  lag  of  the  currents  with  respect  to  the  voltages 
of  the  respective  branches  are  zero,  since  the  branches  are  non- 
reactive. 


(5)  The  currents  in  the  line  wires,  IAL , IBL , and  ICL  are,  of 
course,  equal  to  respectively  and  in  phase  with  the  branch  cur- 
rents IqA , IoB ’ ^IRl  Ioc. 

The  phase  angle  between  EBA  and  IAL  (=/0A)  is,  from  the 


vector  equation  of  the  current,  0BA  = — tan-1 

15.8 
36.4 


-15.8 


= 23°  28'. 


The  minus  sign  is  applied  to  tan  1 ( — 


36.4 

, because  by  conven- 


tion the  phase  angle  between  current  and  voltage  is  positive 
when  measured  from  the  current  to  the  voltage  in  counter-clock- 
wise direction , i.e.  for  a lagging  current.  In  this  case  we  have 
taken  the  voltage  as  the  initial  line,  and  measuring  from  it 
gives  a reversed  angle. 

Moreover,  since  ECB  lags  120°  behind  EBA  or  what  is  the 
same  thing  leads  by  240°,  the  angle  between  I0B  and  ECB  is 


@cb  — 


240°  - tan"1 


-15.8 


— (100  — 36.4)/_ 


= 46°  3'. 


POLYPHASE  CIRCUITS  AND  MEASUREMENT  POWER  377 


In  like  manner  the  phase  angle  is 


6 


AC  — 


120° -tan-1 


f 86.6-15.8Y] 
V 36.4-50  )_ 


= - 19°  9'. 


Here  EAC  lags  240°  behind  EBA , or,  for  simplicity,  in  the  formula 
may  be  counted  as  leading  by  120°. 

It  must  be  understood  that  these  angles  are  the  angles  between 
the  respective  line  currents  and  line  voltages,  and  do  not  deter- 
mine the  power  factor,  which  is  fixed  by  the  phase  relations  of 
the  branch  currents  with  respect  to  the  branch  voltages  and  is 
unity  in  this  case. 

(c)  The  power  in  the  circuit  is  EOAIOA  + E0BI0B  + EocIoc 
= 5460  watts,  since  the  branches  are 
non-reactive. 

A check  can  be  applied  to  the  com- 
putations made  in  finding  the  currents 
by  constructing’  a vector  diagram  as 
in  Fig.  d.  Here,  A'B'  is  equal  and 
parallel  to  IOA,  B'C  is  equal  and 
parallel  to  I0B,  and  O' A'  is  equal  and  parallel  to  Ioc.  Had 
the  triangle  not  closed,  it  would  have  been  an  indication  of 
error,  as  in  such  a case  the  instantaneous  sum  of  the  branch 
currents  could  not  be  zero  at  the  neutral  point  0,  Fig.  a. 


C 

Example  1.  — Figure  d. 


Example  2. — A three-phase  line  (. AlBlOl , Fig.  a ) is  con- 
nected to  a wye  circuit  having  the  following  characteristics : 

In  branch  OA,  R0A  = 
1 ohm  in  parallel  with 
in 


-Al 


X0A  = + 1 ohm ; 
branch  OB , ROB  = 2 
ohms  in  parallel  with 
XOB  = — 1 ohm  ; and  in 
branch  OC. , Roc  = 1 ohm 
(Fig.  a).  Equal  vol- 
tages of  100  volts  are  impressed  on  the  three  phases.  Answer 
the  questions  a,  b , and  c of  Ex.  1. 

The  same  preliminary  or  trial  diagram  as  was  previously 
used  (Ex.  1,  Fig.  5)  may  be  employed  here  for  construct- 
ing the  vector  equation  of  branch  voltages.  They  are  as 
before : 


378 


ALTERNATING  CURRENTS 


Eoa  = « ~jb- 

£0B  = “ (10°  - a)  ~ Jb- 

U0c  = « — 50  4-  j (86.6  — £). 


Example  2. — Figure  b. 

Then 


(a)  To  obtain  the 
vector  expression  for 
branch  currents,  the 
voltage  equations  must 
be  divided  by  the 
respective  branch  im- 
pedances or  multiplied 
by  the  respective  ad- 
mittances. As  a matter 
of  convenience,  multi- 
ply by  the  admittances, 
which  are 

YOA=l-jt; 

1 ob  = T ,7  1 5 
and  Yoc  = 1.* 


Iqa  = («  (1  ~j  1)  = « - b ~j  (a  + J). 


I0B  = («  —100  — jb)Q  +j  1)  = | a — 50  + b +j(  — | b + a — 100). 
Ioc  — (a  — 50  4 j (86.6  — £>))  (1)  = a — 50  +/  (86.6  — J). 


By  placing  the  vertical  and  horizontal  components  respectively 
equal  to  zero,  there  results 

a = 40  and  b = — 5.3, 

and  Fig.  b can  now  be  plotted  to  scale  as  shown. 

The  scalar  values  of  the  branch  voltages  are  : 

Eoa  = V (10)2  4 (5.3)2  = 40.3  volts. 

Eob  = V (60)2  4 (5.3)2  = 60.2  volts. 

Eoc  — V (10)2  4 (92)2  = 92.5  volts. 


* Arts.  74  and  78. 


POLYPHASE  CIRCUITS  AND  MEASUREMENT  POWER  379 


The  currents  may  be  obtained  in  the  same  manner  from  the 
vector  equations  of  currents,  and  are  : 

IOA  = V (40  + 5. 3)2  + (40  — 5. 3)2  = 57. 2 amperes. 

Iob  = ^ (20  — 50  — 5. 3)2  + 4-  40  — loo)  = 67.4  amperes. 

Ioc  = V (40  — 50)2  4-  (86.6  -f-  5.3)2  = 92.5  amperes. 


The  angles  of  lag  between  the  currents  and  voltages  of  the 
branches  are  found  directly  from  the  admittances  and  are  : 

d0A  = — tan-1  — — - = 45°  ; 0OB  = — tan-1  j = — 63°  26' ; 

1 2 

0OC  — — tan-1  ^ = 0°. 


h.  The  currents  in  the  branches  are  the  same  as  in  the  lines 
to  which  they  are  connected. 
c.  The  power  in  the  circuit  is 

1-OA  OA  ^ OA  4"  I()B  -® OB  C0S  ^ OB  4“  ^ OC  ~^OC  C0S  ^ OC 

= 12,000  watts. 


The  way  in  which  the  branch  voltages  are  influenced  by  the 
power  factors  of  the  branches,  as  well  as  by  the  magnitudes  of 
the  branch  currents,  is  clearly  seen  in  this  example. 


Example  3.  — A 

neutral  wire  is  con- 
nected to  the  point  0 
in  the  wye  of  Ex.  2, 
Fig.  a,  and  is  main- 
tained at  fixed  equal 
voltages,  120°  apart, 
relatively  to  the  re- 
spective potentials  of 
the  points  A,  B,  and 
C.  Answer  questions 
a and  c of  Ex.  1,  and 
calculate  the  current 
in  the  neutral  wire. 

(a)  In  this  case  the 
point  0 in  the  trial 


-Y 

Example  3.  — Figure  a. 


380 


ALTERNATING  CURRENTS 


vector  diagram  of  Ex.  2,  Fig.  a,  is  fixed  at  the  center  of  the 
triangle  by  the  effect  of  the  neutral  wire ; therefore,  the  effect- 
ive voltages  on  the  branches  are  equal  numerically  to  each 

other  and  of  value  = 57.7  volts.  Figure  a shows  the  rela- 
V3 

tions  of  the  voltages.  The  vector  currents  in  the  branches, 
each  referred  to  its  branch  voltage  as  the  initial  line,  also  shown 
in  Fig.  a,  are  as  follows  : 

i0A  = Ka  Yoa  = 57.7  (1  -j  1)  = 57.7  -j  57.7. 

i0B  = YJubYob  = 57.7  (1  +j  1)  = 28.9  +j  57.7. 

loc  — E0CY0C  = 57.7  (1  +j  0)  = 57.7  +j  0. 

The  scalar  values  and  the  angles  of  lag  are : 

I0A—  V(57.7)2  -t-  (57. 7)2  = 81.7  amperes.  0OA  = 45°. 

lOB  = V (28. 9)2  -f-  (57. 7)2  = 64.6  amperes.  d0B  = — 63°  26'. 

Ioc  = 57.7  amperes.  6oc  = 0. 

The  neutral  current  ION  must  have  such  a value  that  the  alge- 
braic sum  of  instantaneous  currents  meeting  at  0 will  be  equal 
to  zero,  but  each  of  the  vector  equations  of  currents  just  given 
has  its  components  referred  to  its  own  branch  voltage  for  the 
horizontal  axis.  These  are  120°  apart;  the  coordinates  of  the 
current  vectors  must,  therefore,  be  changed  so  as  to  refer  to  a 
single  pair  of  axes  before  they  can  be  added.  We  will  take  as 
the  axes  for  this  OX  and  OY,  where  OX  coincides  with  the 
line  voltage  EBA.  It  conduces  to  simplicity  to  use  one  of  the 
line  voltages  as  the  initial  line  in  this  manner. 

Then,  b}r  using  the  proper  direction  coefficients, 

I'oa  = (57.7  — j 57.7)  cjs*  (—  30)  = (57.7  —j  57.7)(.866  —j  .5) 
= 21.1  — j 78.8. 

I'OB  = (28.9  + j 57.7)  cjs  ( — 150°)  = 3.87  — j 64.4. 

I'oc  = (57.7  +j  0)  cjs  (-  270°)  = 0 + j 57.7. 

But  I'oa.  + 1' OB  + Y oc  + I'ox  = 0 5 


* Art.  76. 


POLYPHASE  CIRCUITS  AND  MEASUREMENT  POWER  381 


hence  substituting  the  right-hand  sides  of  the  above  equations 
in  this  and  transposing  gives, 

I' ON  = — 25  + / 85.5, 

with  a scalar  value  of  lON  = 89.1  amperes,  and  an  angle  with 
reference  to  the  direction  axis  OX  of  0'ON  = 106°  18'. 

(c)  The  power  may  be  found  by  taking  the  product  of  the 
current,  voltage,  and  cosine  of  angle  of  lag  of  each  branch  and 
adding  the  three  products  together. 

This  and  the  previous  problems  show  clearly  the  advantage 
of  using  a neutral  lead  wire  on  a star  load  for  the  purpose  of 
keeping  the  voltages  in  the  branches  as  nearly  as  possible  equal, 
and  to  prevent  severe  insulation  strains  and  poor  service. 

The  check  may  be  obtained  as  in  Ex.  1 by  drawing  the 
vector  polygons  of  voltages  or  currents.  The  vector  polygon 
of  line  voltages  is  partly  dotted-in,  in  Fig.  a.  In  making  the 
check  polygon  of  current  vectors,  the  currents  should  be  drawn 
parallel  to  and  of  equal  magnitude  to  their  vectors  in  the  phase 
diagram  ; and  should,  to  avoid  confusion,  be  preferably  taken  in 
the  order  in  which  they  appear  on  that  diagram  when  following 
it  around  in  a clockwise  direction.  Thus,  to  IOA  add  IOB,  then 
add  I0N , and  then  Ioc.  In  order  that  the  solution  may  be 
correct,  it  is  evidently  necessary  for  the  neutral  current  to 
close  an  open  polygon  of  branch  currents. 

Example  4.  — A star  circuit  having  the  same  impedances  in  its 
branches  and  the  same  arrangement  as  in  Ex.  2 has  impressed 
voltages  as  follows  : EAB  — 141.4  ; EBC  = 100  ; ECA  = 100. 

Answer  the  questions  of  Ex.  2. 

Suggestion  : Lay  out  the  vector  polygon  of  voltages  as  in 
Ex.  2,  making  EBA  horizontal.  Assume  a point  O'  and  solve 
for  the  position  of  the  neutral  point  as  before.  Lay  out  the 
phase  diagram  and  proceed  as  in  Ex.  2,  using  care  to  plot 
the  current  phase  angles  correctly.  The  problem  is  in  essence 
the  same  and  as  simple  as  that  of  Ex.  2,  and  should  be  solved 
without  difficulty.  These  same  unbalanced  voltages  are  used 
in  Ex.  5,  which  has  to  do  with  a delta  system. 

Example  5.  — The  branch-load  impedances  of  Ex.  2 are  con- 
nected in  the  form  of  a delta  as  shown  in  Fig.  a\  141.4  volts 


382 


ALTERNATING  CURRENTS 


are  impressed  on  the  phase  AB  and  100  volts  are  impressed 
upon  each  of  the  other  phases. 

(a)  What  is  the  current  and  angle  of  lag  in  each  branch? 

( b ) What  is  the  current  in  each  line  and  its  lag  angle  with 

reference  to  one  of 
the  line  voltages  ? 

(a)  First  the  vec- 
tor polygon  of  vol- 
tages is  made  and  then 
the  phase  diagram  by 
drawing  the  voltage 
vectors  parallel  to  those  of  the  vector  polygon  ; or  as  the  angles 
of  the  triangle  and  their  supplements  can  readily  be  obtained 
by  the  elementary 
formulas  of  trigo- 
nometry, the  vector 
polygon  may  be 
omitted.  Using  the 
latter  method,  we 
find  that  E, 


CB 


lags 


E, 


BA1 

asrs 


Eb 


voltages  are 


135°  behind 
and  that  EA 
225°  behind 
These 

laid  out  in  Fig.  b 
with  Eba  on  the 
initial  line  OX , as 
heretofore.  The 
vector  equation  of 
each  branch  current, 
with  components 
taken  with  reference  to  its  particular  branch  voltage,  is 
obtained  by  multiplying  the  branch  voltage  by  the  correspond- 
ing branch  admittance  ; so  that, 


Example  5. — Figure  b. 


IBA  = 141.4  -j  141.4. 


ICB  = 50  +j  100. 

4 = 100. 


POLYPHASE  CIRCUITS  AND  MEASUREMENT  POWER  383 


From  this  the  scalar  values  of  the  currents  in  amperes  and 
the  angles  of  lag  are  found  to  be : 

IBA  = 200  amperes.  6BA  = 45°. 

ICB  = 111.8  amperes.  0CB  = —63°  26'. 

IAC  = 100  amperes.  dAC  = 0°. 

(5)  The  currents  in  the  three  line  conductors  may  be  found 
by  combining  the  branch  currents  graphically  as  shown  in  Fig. 
b,  or  they  may  be  determined  more  accurately  by  adding  their 
vectors  geometrically.  Thus,  referring  each  of  the  branch  cur- 
rents to  the  axes  OX,  OY,  there  results, 

* ba  = Iba  <ds  (0°)  = 141-4  -j  141.4. 

1'cb  = Tcb  cjs  ( - 135°)  = 35.3  -j  106. 

I' AC  = Iac  cjs  (-225°)=  -70.7  + ,?  70.7. 

Remembering  that,  by  the  convention  adopted  in  this  book, 
current  at  a junction  point  is  considered  positive  when  it  flows 
toward  the  point,  the  two  currents  meeting  at  a corner  of  the 
mesh  may  be  added,  giving  the  following  results : 

I'AL  = I' BA  + I'CA  = I' BA  + i-I’Ac)  = 212-1  -j  212.1. 
Likewise,  T BL  = I'CB  + T AB  = - 106. 1 +j  35.4, 

and  Tcl  = I'AC  + I’BC  = - 106  +/ 176.7. 

The  scalar  values  of  these  currents  from  these  equations  are : 
IAL  = 300  amperes  ; IBL  = 112  amperes  ; ICL  — 206  amperes. 

These  currents  have  the  following  phase  relations  with  the 
line  voltages  : IAL  lags  behind  XBA  by  45°  ; IBL  lags  behind  ECB 
by  63°  26' ; and  ICL  lags  behind  EAC  by  14°  5'.  These  angles 
are  obtained  from  the  vector  equations  of  current  representing 
V T'  iiiid  V 

-L  AL'  -L  BL ' CL* 

Example  6. — The  wye  circuits  of  Ex.  1 and  Ex.  2 are  placed 
in  parallel  on  a set  of  100-volt,  three-phase  mains  of  the  fre- 
quency assumed  in  those  examples.  What  total  current  flows 
in  the  mains? 


384 


ALTERNATING  CURRENTS 


All  that  is  necessary  is  to  geometrically  add  the  currents 
already  determined  in  those  problems.  (A  line  is  drawn  under 
the  subscripts  of  the  symbols  from  Ex.  1 to  distinguish  them 
from  those  of  Ex.  2.)  Thus, 

IgA  + Ioa  = (36.4  - j 15.8)  + (45.3-  j 34.7) 

= 81.7  — j 50.5, 

from  which  IAL  = 96  amperes.  The  total  currents  in  the  other 
lead  wires  can  be  obtained  in  the  same  manner. 

Example  7.— A load  of  10,000  watts  is  absorbed  in  a delta 
three-phase  system  in  the  following  way : in  the  branch  AB , 
4000  watts  at  90  per  cent  power  factor,  current  leading;  in  the 
branches  BC  and  CA,  3000  watts  each  at  80  per  cent  power 


factor,  current  lagging.  What  are  the  currents  in  the  branches 
and  in  the  lines,  the  voltage  between  A and  B being  141.4 
volts,  between  B and  (7,  100  volts,  and  between  C and  A . 100 
volts  ? 

From  the  data  given  we  have, 

Iba  = I4i°4°x  9 = 31,4  amPeres-  9 ba  = - 25°  50'. 

ICB  = = 37.5  amperes.  0CB  = 36°  52'. 

I ag  = "(^°g  = 37.5  amperes.  0AC  = 36°  52'. 

The  combined  currents  can  be  obtained  graphically  as  shown 
in  Fig.  b , but  more  accurately  by  adding  their  vectors  referred 
to  a common  axis,  OX , as  has  been  done  heretofore,  care  being 


POLYPHASE  CIRCUITS  AND  MEASUREMENT  POWER  385 

used  to  give  the  dividing  currents  at  a branch  junction  the 
correct  sign.  The  vector  equations  are : 

I'BA  = Iba  = 28-3  +3  13-7- 

I'CB  = ICB  e-7(36°a2-+i35°;=  _37.i  _/  5.30. 

l'AC  = /J  € -| (36°  5*+225P)  = - 5.30  + j 37. 1. 

The  exponential  form  of  the  complex  expression  is  used  here 
as  a matter  of  convenience  in  computation,  as  might  have  beeu 
done  also  in  earlier  examples.  From  these  expressions  of  the 
vector-branch  currents  we  obtain 


Ial  = I'ba  + I'ca  = 33-3  — j 23.4. 
IBL  = I'cb  + I'ab  = - 65.4  -j  19.0, 
ICL  = I’ AC  + I'bc  = 31-8  +j  42.4. 


The  scalar  values  of  the  line  currents  then  are : 

7^  = 40.8;  7Bi  = 68.1;  and  ICL  = 53  amperes. 

It  should  be  noted  that  where  the  voltages  or  impedances  are 
unequal,  as  in  this  case  and  in  Ex.  5,  the  delta  branch  currents 
do  not  necessarily  form  a closed  vector  polygon,  though  the 
three  line  currents  do. 


2c 


380 


ALTERNATING  CURRENTS 


Example  8. — A two-phase  receiver  having  a common  return 
wire  (Fig.  a ) has  the  following  characteristics : In  OA,  R = 2 

B in  parallel  with 

-X"  = — 1 ; and 


B 


R=i;X  = i 0 


Ebo=  100 

1 


X-i  R=2 


■0L 


F =141.4 

Q AO 

i 


in 

OB , R = 1 in  series 
with  X = 1.  Be- 
tween the  line  wires 
Al  and  0L,  and  BL 
and  0L,  respectively, 
141.4  and  100  volts  are  impressed.  What  are  the  currents  in 
the  three  line  wires  and  the  voltage  between  the  lines  AL  and  BL 
when  Eoa  leads  Bob  by  90°. 


Example  8.  — Figure  a. 


■Al 


Yoa  = 2 +i  1 and  Y0 


X _ n X 
2 «/  2’ 


and  the  vectors  of  current  referred  to  their  respective  voltages 

lot  = Eoa  Yoa  = 70.7  +j  141.4, 

I0b  — Bob  5 ob  — 50  j 50  ; 

from  which,  IOA  = 158.1  amperes  and  dOA  = — 63°  26';  and 
I0B  =70.7  amperes  and  0OB  = 45°. 

Referring  the  vectors  IOA  and  I0B  to  the  axes  OX ’,  OY , 

I'OA  = IoAej^^  = 10.1+jUlA, 
and  I'OB  = I0Be~  j (45° + 9°0)  = - 50  - j 50. 


Adding  these  vectors  gives, 

IlO  — I'oA  + I'oB  — 20.7  + J 91.4. 

The  scalar  value  of  IL0  is  93.7  amperes. 

The  voltage  between  AL  and  BL  is 

Bba  — Ebo  + Boa  — 141.4  + j 100, 
or  Bba  = 173  volts  at  an  angle  leading  BOA  by  35°  16'. 

Prob.  1.  A three-phase  wye  circuit  has  in  series  in  each 
branch  a resistance  of  two  ohms  and  an  inductive  reactance  of 
one  ohm  When  a line  voltage  of  100  volts  is  impressed  on 


POLYPHASE  CIRCUITS  AND  MEASUREMENT  POWER  387 


each  phase,  what  current  flows  in  each  branch,  what  are  the 
branch  voltage  and  angle  of  lag,  and  what  are  the  power  ex- 
pended in  the  circuit  and  the  power  factor  ? 

(In  these  problems  lay  out  the  results  of  computations  in 
phase  diagrams.) 

Prob.  2.  If  the  power  of  10,000  watts  is  expended  in  the 
wye  system  of  Prob.  1,  what  must  be  the  voltage  across  the 
mains,  and  the  cur- 
rents in  the  mains 
and  branches? 

Prob.  3.  A three- 
phase  delta  circuit 
has  a resistance  of 
one  ohm  and  capac- 
ity reactance  of  one 
ohm  in  series  in  each 
branch.  When  500 
line  volts  are  im- 
pressed on  each 
phase,  how  much 
current  flows  in  each 
branch  and  line  ? 

How  much  power  is 
expended  in  the  cir- 
cuit, and  what  is  the 
power  factor  ? 

Prob.  4.  When  25 
Ivw.  are  expended  in 
the  delta  of  Prob.  3,  what  currents  flow  in  the  branches  and 
line  conductors  and  what  are  the  line  voltages  ? 

Prob.  5.  If  two  wye  circuits,  having  the  characteristics  and 
being  under  the  conditions  named  in  Prob.  1,  are  connected  in 
parallel  to  the  same  line  wires,  will  current  flow  through  a con- 
ductor joining  their  neutral  points  ? Prove  the  answer. 

Prob.  6.  A three-phase  transmission  line  receives  from  a 
generating  plant  1000  Kw.,  the  line  voltage  being  5000  volts 
in  each  phase.  How  much  current  flows  in  each  wii’e  if  the 
circuit  is  balanced  and  of  90  per  cent  power  factor? 

Prob.  7.  The  generator  of  Prob.  6 is  wye-connected.  What 
voltage  must  be  produced  by  the  coils  of  each  phase? 


Y 


Example  8. — Figure  b. 


388 


ALTERNATING  CURRENTS 


Prob.  8.  A balanced  load  of  500  K\v.  is  transmitted  over  a 
two-phase  line  of  three  wires,  at  a voltage  of  2000  volts  per 
phase,  to  induction  motors  having  an  average  power  factor  of 
85  per  cent.  What  current  flows  in  each  of  the  three  wires 
and  what  is  the  voltage  between  the  two  independent  wires? 

Prob.  9.  A three-phase  wye  circuit  has  resistance  of  one 
ohm  and  capacity  reactance  of  .25  ohms  in  series  in  each 
branch.  The  line  voltages  EAB,  EBC,  and  ECA,  respectively, 
are  200,  220,  and  240  volts.  What  are  the  values  of  current 
and  voltage  in  each  branch,  and  what  are  the  angles  of  lag  of 
the  currents  with  respect  to  the  corresponding  branch  voltages? 

Prob.  10.  A three-phase  wye  circuit  has  the  following  vec- 
tor impedances  in  its  branches:  Z0A  — l-\-jl\  ZOB=\-\-j\\ 
and  Zoc  = \—j  1.  What  are  the  branch  voltages,  currents,  and 
angles  of  lag  when  a line  voltage  of  500  volts  is  maintained  on 
each  phase?  What  power  is  absorbed  by  the  circuit  ? 

Prob.  11.  The  impedances  of  Prob.  10  are  connected  in 
delta  to  the  same  mains.  What  are  the  branch  currents  and 
angles  of  lag,  and  what  are  the  line  currents?  What  power  is 
absorbed  by  the  circuit  ? 

Prob.  12.  A neutral  wire  is  connected  to  the  wye  junction 
of  Prob.  10  without  changing  the  other  conditions,  except  that, 
by  proper  regulators,  the  voltages  between  the  neutral  point 
and  the  line  conductors  are  maintained  numerically  equal. 
What  current  flows  in  the  neutral  wire,  and  what  power  is 
absorbed  by  the  circuit  ? 

Prob.  13.  A transmission  line  supplies  a three-phase  wye 
circuit  with  100  Kw.  in  the  proportion  of  50  Kw.  at  85  per  cent 
capacity  power  factor  in  one  branch  and  25  Kw.  at  unity  power 
factor  in  each  of  the  other  branches.  The  line  voltage  of  each 
phase  is  240  volts.  What  are  the  values  of  the  branch  currents, 
voltages,  angles  of  lag,  kilowatts,  and  kilovolt-amperes,  what 
are  the  line  currents,  and  what  are  the  total  kilovolt-amperes 
of  the  circuit? 

Prob.  14.  A transmission  line  supplies  the  three  branches 
of  a delta  circuit  with  1000  Kw.  in  the  proportion  of  250,  350, 
and  400  Kw.,  the  voltage  and  power  factor  for  each  branch 
being  2000  volts  and  90  per  cent  power  factor.  What  are  the 
branch  currents,  the  line  currents,  the  circuit  power,  and  the 
circuit  kilovolt-amperes  ? 


POLYPHASE  CIRCUITS  AND  MEASUREMENT  POWER  389 


Prob.  15.  A three-phase  transmission  line  of  2000  line  volts 
on  each  phase  supplies  the  following  circuit : a three-phase 

delta  circuit  taking  20  Kva.*  in  each  branch  at  80  per  cent 
lagging  power  factor ; a three-phase  wye  circuit  taking  50  Kva. 
in  each  branch  at  90  per  cent  leading  power  factor ; and  a 
single-phase  load  from  one  phase,  of  25  Kw.  at  unity  power 
factor.  How  much  current  flows  in  each  line  wire  ? 

Prob.  16.  A balanced  load  of  100  Kw.  at  100  per  cent  power 
factor  is  absorbed  by  a six-phase  mesh-connected  receiver.  The 
voltage  between  line  wires  is  250  volts.  What  is  the  current 
per  line  wire? 

Prob.  17.  What  would  the  branch  voltages  and  line  currents 
be  if  the  load  of  Prob.  16,  at  the  same  line  voltages,  was  absorbed 
by  a six-phase  star  circuit  ? 

104.  Measurement  of  Power  in  Two-  and  Three-phase  Cir- 
cuits. — The  principles  underlying  the  methods  of  measuring 
power  in  polyphase  circuits  differ  in  no  respect  from  those 
already  deduced  in  relation  to  single-phase  circuits,!  but  it  is 
desirable  to  apply  them  in  such  a way  as  to  reduce  the  number 
of  necessary  readings  to  a minimum.  For  satisfactory  measure- 
ments, non-reactive  wattmeters  are  of  essential  importance. 

A.  Two-phase  Systems 

A 1.  Independent  Circuits.  In  a two-phase  system  with 
separate  circuits,  independent  wattmeter  readings  are  taken  in 
each  circuit  and  the  total  power  is  the  sum  of  the  readings. 
One  wattmeter  placed  in  each  circuit,  as  shown  in  Fig.  231, 
from  which  simultaneous  readings  are  taken,  is  the  best  arrange- 
ment ; but  if  two  wattmeters  are  not  to  be  had,  one  may  be 
inserted  successively  in  the  two  circuits,  and  the  sum  of  the 
readings  is  equal  to  the  power  in  the  system,  provided  the  load 
does  not  vary  while  the  readings  are  being  taken.  If  the  circuit 
is  perfectly  balanced,  twice  the  reading  of  a wattmeter  in  one 
circuit  is  equal  to  the  power,  but  this  is  a condition  which  can- 
not be  relied  upon. 

A 2.  Circuits  with  Common  Return.  Two  wattmeters  may 
here  be  used,  one  for  each  circuit,  connected  in  the  way  shown 

* Kva.  is  an  abbreviation  for  kilovolt-amperes.  t Art.  99. 


390 


ALTERNATING  CURRENTS 


W 


Fig.  231.  — Measurement  of  Power  in  Two-Pliase  System  with  Separate  Circuits. 

in  Fig.  232.  The  arrangement  shown  in  Fig.  233  is  equivalent 
to  a single  wattmeter  connected  as  in  Fig.  234,  and  is  only 
correct  for  a system  in  exact  balance.  When  the  single  watt- 


w 


Fig.  232.  — Measurement  of  Power  in  Two-phase  System  with  Common  Return. 

meter  is  used  in  a balanced  system,  the  current  coil  is  placed  in 
the  common  wire,  and  a reading  is  taken  with  the  free  end  of 
the  voltage  coil  connected  to  one  outside  wire.  The  voltage 
coil  terminal  is  then  quickly  transferred  to  the  other  outside 


POLYPHASE  CIRCUITS  AND  MEASUREMENT  POWER  391 


Fig.  233.  — Measurement  of  Power  in  Two-phase  System  in  Exact  Balance. 


wire  and  a new  reading  taken.  The  condition  of  exact  balance 
is  not  to  be  relied  upon,  so  that  the  arrangement  of  Fig.  232 


Fig.  234. — Equivalent  Connection  for  Fig.  233,  using  One  Wattmeter. 

must  ordinarily  be  used.  The  sum  of  the  readings  of  the  two 
wattmeters  then  gives  the  power  in  the  system. 

B.  Three-phase  Systems 

B 1.  Three  Wattmeters.  a.  If  the  power  delivered  by  a 
generator,  or  absorbed  by  a motor  or  other  device,  which  is 
connected  in  wye  fashion,  is  to  be  measured,  three  wattmeters 
may  be  used,  connected  as  shown  in  Fig.  235,  provided  the 
common  or  neutral  point  is  accessible.  It  is  evident  that  each 


392 


ALTERNATING  CURRENTS 


W 


Fig.  235.  — Measurement  of  Power  in  Three- 
phase  System,  Wye  Connected.  Three 
Wattmeters  used. 


wattmeter  measures  the  power  in  one  branch  so  that  the  sum 
of  the  readings  gives  the  power  in  the  system.  If  the  system 

is  exactly  balanced,  three 
times  the  reading  of  one 
wattmeter  gives  the  power. 

b.  If  the  devices  are  con- 
nected delta  fashion,  three 
wattmeters  may  still  be 
used,  provided  the  current 
coils  of  the  wattmeters  can 
be  inserted  directly  in  the 
branch  circuits  as  shown  in 
Fig.  236.  The  power  in 
the  circuit  is  equal  to  the 
sum  of  the  three  wattmeter 
readings,  and  if  the  circuit 
is  exactly  balanced,  three  times  the  reading  of  one  wattmeter 
gives  the  power. 

c.  When  it  is  impossible  to  insert  the  wattmeters  in  the  coil 
circuits  of  a device  with  delta  connection,  the  three-wattmeter 
method  may  still  be  used  A 
by  the  creation  of  an 
artificial  neutral  point  as 
shown  in  Fig.  237.  For 
this  purpose,  three  equal 
non-reactive  resistances 
are  connected  together 
at  one  end  of  each,  and  the 
other  ends  are  connected 
to  the  respective  corners 
of  the  mesh  circuit.  In  a 
balanced  circuit,  the  vol- 
tage between  the  neutral 
point  and  either  corner 
E 


Fig.  236.  — Measurement  of  Power  in  Three- 
phase  System,  Delta  Connected.  Three  Watt- 
meters used. 


is  equal  to  — where  E 
V3 

is  the  voltage  of  one  branch,  and  the  phase  of  this  voltage  is  0 
degrees  in  advance  of  the  current  entering  the  corner.  A watt- 
meter with  its  current  coil  inserted  in  the  circuit  wire  leading  to 
the  corner  carries  a current  equal  to  V3  /,  where  I is  the  branch 


POLYPHASE  CIRCUITS  AND  MEASUREMENT  POWER  393 


current,  and  if  the  free  end  of  the  voltage  coil  is  connected 
to  the  neutral  point,  the  power  reading  of  the  wattmeter  is 

w 


V3  IE  cos  9 

vF- 


= IE  cos  9, 


Fig.  237.  — Measurement  of  Power  in 
Three-phase  System,  Delta  Con- 
nected, with  Artificial  Neutral. 


which  is  the  power  in  the 
branch.  Care  must  be  taken 
that  the  resistances  of  the  watt- 
meter voltage  coils  are  so  large 
compared  with  the  three  auxil- 
iary resistances  that  connecting 
them  in  circuit  does  not  disturb 
the  voltage  of  the  neutral  point. 

d.  Tf  the  free  ends  of  the  three  voltage  coils  are  connected 
directly  together,  the  sum  of  the  instrument  readings  gives 
the  total  power,  although  the  readings  may  not  be  individually 
significant  unless  the  resistances  of  the  voltage  coils  are  exactly 
equal. 

These  methods  are  independent  of  the  condition  of  balance  in 
the  system  or  the  current  lag. 

B 2.  Two  Wattmeters.  The  algebraic  sum  of  the  readings 
of  two  wattmeters  inserted  in  a three-phase  circuit  as  shown  in 
yj  Fig.  238,  gives  the  power  in  the 

system  with  entire  independence 
of  the  balance  of  the  system  or 
current  lag.  When  the  current 
lag  in  the  circuit  is  less  than  60°, 
i.e.  the  power  factor  is  greater 
than  .50,  the  arithmetical  sum 
of  the  readings  is  equal  to  the 
power  in  the  circuit  ; but  if  the 
lag  is  greater  than  60°  (the  power 
factor  is  less  than  .50),  the  rela- 
tion of  the  currents  in  the  current  and  voltage  coils  of  one  of 
the  wattmeters  causes  it  to  have  a negative  reading,  and  the 
arithmetical  difference  of  the  readings  of  the  two  instruments 
gives  the  power.  There  is  some  difficulty  in  distinguishing 
which  condition  exists  in  some  cases,  especially  when  the  power 
absorbed  by  partially  loaded  induction  motors,  in  which  the 
power  factor  is  low,  is  under  measurement.  As  a general  rule, 


Fig.  238.  — Measurement  of  Power  in 
Three-phase  System.  Two  Watt- 
meters used. 


394 


ALTERNATING  CURRENTS 


if  the  conditions  do  not  make  the  case  evident,  the  truth  mav 
be  discovered  by  interchanging  the  positions  of  the  instruments 
without  altering  the  relative  connections  of  their  main  and 
voltage  coils.  If  the  deflections  of  both  needles  are  reversed, 
the  difference  of  the  original  readings  represents  the  power, 
but  if  the  deflections  are  in  the  same  direction  as  before,  the 
sum  of  the  readings  is  correct.  The  proof  of  this  theorem  is 
given  in  Art.  105. 

A double  wattmeter,  consisting  of  two  movable  coils  on  one 
spindle  and  two  fixed  coils,  can  be  used  in  measuring  power 
by  the  two-wattmeter  method.  Such  an  instrument  of  itself 
sums  up  the  double  reading  algebraically,  and  a single  reading 
gives  the  power.  Integrating  wattmeters  based  upon  this  prin- 
ciple are  very  useful  in  the  commercial  sale  of  power  from  two- 
phase  and  three-phase  circuits. 

B 3.  One  Wattmeter.  In  a balanced  circuit  one  wattmeter 
may  be  very  conveniently  used  by  connecting  the  current  coil 

in  one  wire  and  connecting  the 
free  terminal  of  the  voltage  coil 
alternately  to  the  other  two  leads 
(Fig.  239),  when  the  sum  of 
the  readings  gives  the  power. 
For,  the  power  reading  of  the 
wattmeter  in  its  first  position  is 
V3  IE  cos  (0  + 30°),  and  in  its 
second  position, 

V3  IE  cos  (0  — 30°), 

and  the  sum  of  the  readings, 

V3  IE  \ cos  (0  + 30°)  + cos  (0  - 30°)}  = 3 IE  cos  0, 

where  I E.  and  0 are  the  current,  voltage,  and  lag  in  a branch ; 
but  IE  cos  0 is  the  power  in  one  branch  and  3 IE  cos  0 is  the  total 
power  of  the  three  branches,  hence  the  one  wattmeter  gives  correct 
indications  provided  0 is  the  same  for  all  the  branches,  and  the  load 
is  uniformly  distributed.  A wattmeter  having  two  independent 
voltage  coils  could  be  used  as  a direct  reading  instrument  for 
this  purpose.  A similar  wattmeter  could  also  be  used  in  one- 
wattmeter  measurements  of  power  in  two-phase  circuits. 

Thus,  when  one  wattmeter  is  used  as  explained  above  in  a 
balanced  three-phase  circuit,  the  sum  of  the  two  readings  gives 


Fig.  239.  — Measurement  of  Power  in 
Three-phase  Balanced  System.  One 
Wattmeter  used. 


POLYPHASE  CIRCUITS  AND  MEASUREMENT  POWER  395 


3 El  cos  9.  In  like  manner  it  may  be  shown  that  the  difference 
of  the  same  two  readings  gives  V3  El  sin  6.  From  which  it 

follows  that  ,,,  nn 

tan0  = V3^-^, 
u\  + %<2 

where  wx  and  w2  are  the  two  readings  of  the  wattmeter.  This 
relation  is  sometimes  convenient  for  obtaining  the  value  of  9 and 
thus  the  power  factor  of  a balanced  three-phase  circuit.  It  may 
also  be  shown  by  a similar  process  of  reasoning  that,  when  the 
circuit  is  balanced,  two  wattmeters  connected  in  the  manner 
illustrated  in  Fig.  238  give  readings  which  may  also  be  used  in 
this  formula. 

Wye  Box.  The  power  in  a balanced  three-phase  system  may 
also  be  measured  by  using  one  wattmeter  and  what  is  called  a 
Y-box.  This  consists  of  two  equal  non-reactive  resistance  coils 
with  one  end  of  each  connected  to  a common  binding  post  and 
the  other  end  of  each  connected  to  an  individual  binding  post. 
If  each  coil  is  of  resistance  equal  to  the  resistance  of  the  voltage 
coil  of  a wattmeter  with  which  it  is  designed  to  be  used,  insert- 
ing the  wattmeter  current  coil  in  one  leg  of  the  circuit,  connect- 
ing the  individual  binding  posts  of  the  wye  box  respectively  to 
the  other  legs  of  the  circuit,  and  connecting  the  free  end  of  the 
wattmeter  voltage  coil  to  the  common  binding  post  of  the  wye 
box,  affords  an  arrangement  equivalent  to  one  wattmeter  of 
Fig.  240.  The  power  for  the 
balanced  system  is  three  times 
the  wattmeter  reading.  If  the 
arms  of  the  wye  box  are  not  each 
equal  in  resistance  to  the  voltage 
coil  of  the  wattmeter,  a correc- 
tion must  be  made. 

105.  Measurement  of  Power  in 
Any  Polyphase  Circuit.  — In  the 

case  of  any  polyphase  system  of  n phases  and  n conductors,  it 
was  originally  proved  by  Blondel  that  the  power  in  the  circuit 
may  be  measured  by  n — 1 wattmeters.  Supposing  A , B,  G , D, 
etc.,  are  points  where  the  n conductors  of  a polyphase  supply 
circuit  connect  to  the  circuits  under  test,  then,  as  has  been 
already  proved,*  1i  = 0,  if  i represents  the  instantaneous  cur- 


Fig.  240. — -Measurement  of  Power 
in.  Three-phase  System.  One  Watt- 
meter used  with  Non-reactive  Wye- 
box. 


* Art.  100. 


396 


ALTERNATING  CURRENTS 


rent  in  any  conductor.  The  power  being  supplied  through  the 

A conductor  at  any  instant  is  equal  to  c^h-  va  = iava,  where  qa  is 

the  quantity  of  electricity  brought  to  A during  a time  dt,  and 
va  is  the  absolute  electrical  potential  of  A. 

The  average  power  transferred  through  conductor  A during 

1 

a complete  period  is  — j iavadt , and  the  total  power  in  the 

T Jo 


circuit,  P 


= 2i  r 

TJo 


ivdt. 


The  absolute  potentials  of  the  points  are  inconvenient  to 
measure,  and  it  is  desirable  to  introduce  into  the  formula  the 
difference  between  the  potentials  at  the  points  and  some  fixed 
point  of  potential  v' . Since  = 0,  we  also  have  = 0,  and 
may  therefore  be  directly  inserted  in  the  formula  without 
destroying  the  equality,  or 


P =1  ]pfT(iv  ~ iv')  dt  = 2 - v') 


dt. 


Writings  for  v — v'  (the  instantaneous  difference  of  potential 
between  the  fixed  point  and  any  given  point  in  the  system)  gives 

P = 2 \ Ciedt. 

TJ0 

The  fixed  point  may  be  taken  at  one  of  the  corners  of  the  cir- 
cuit, A for  instance,  since  SaV  = 0 holds  equally  for  it,  and  the 
power  formula  becomes 

P = 2’^J'  T ijacdt  + etc.  + tneardt , 

or  P — IbEab  cos  Q'  + IcEac  cos  6"  + etc. 


where  0\  6 ",  etc.,  are  the  differences  of  phase  between  the 
respective  line  currents  and  the  corresponding  voltages  to 
point  A.  The  terms  on  the  right  of  the  equation  are  the 
familiar  forms  representing  the  power  readings  of  a wattmeter, 
so  that  if  n — 1 wattmeters  are  inserted  in  circuit  with  their 
current  coils  respectively  in  the  n — 1 conductors,  B.  C , Z),  etc., 
and  the  free  ends  of  their  voltage  coils  all  connected  to  ^4.  the 
algebraic  sum  of  their  readings  is  equal  to  the  power  in  the 
system. 


POLYPHASE  CIRCUITS  AND  MEASUREMENT  POWER  397 


When  the  circuit  includes  three  phases  only,  the  formula  is 
P = IbEob  cos  O'  + IcEac  COS0",  and  only  two  wattmeters,  con- 
nected as  in  Fig.  238,  are  required  to  give  the  power  in  the 
circuit,  but  due  regard  must  be  had  to  the  relative  signs  of 
cos  O'  and  cos  0" . 

From  the  relative  phases  of  the  currents  Ib  and  Ic  and  vol- 
tages Eab  and  Eae  it  is  easy  to  see  that  in  a balanced  three-phase 
system  O'  = 0X  — 30°,  and  also  that  0"  = d2  + 30°,  and  therefore 

P = IbEab  cos  (0X  - 30)°  + IcEac  cos  (02  + 30°), 

in  which  0l  and  d2  are  the  angles  of  lag  of  the  branch  currents, 
and  the  subscripts  indicate  the  respective  branches.  The  for- 
mula shows  that  the  first  term  at  the  right,  which  represents  the 
reading  of  one  wattmeter,  is  positive  within  the  limits  0l  = + 90° 
and  dj  = — 60°,  and  that  its  value  is  negative  between  the  limits 
0l  = —60°  and  6l  = — 90°.  The  second  term,  which  represents 
the  reading  of  the  second  wattmeter,  is  positive  between  the 
limits  — 90°  and  + 60°  and  negative  between  + 60°  and  + 90°. 
Consequently,  if  the  current  lags  equally  in  the  circuits , i.e. 
dj  = : Both  wattmeters  have  a positive  reading  and  the  power 

in  the  circuit  is  the  sum  of  the  readings,  for  angles  of  lag  between 
-f-  60°  and  — 60°.  It'  the  angle  of  lag  is  + 60°  (the  current  lags 
behind  the  voltage),  the  second  wattmeter  reading  is  zero,  and 
the  power  in  the  circuit  is 
equal  to  the  reading  of  the 
first  wattmeter.  If  the  lag  is 
greater  than  + 60°,  the  read- 
ing of  the  first  wattmeter  is 
positive  and  the  second  is 
negative,  and  the  power  in 
the  circuit  is  equal  to  the  dif- 
ference of  the  two  readings. 

Again,  if  the  angle  of  lag  is 
— 60°  (the  current  leads  the 
voltage),  the  first  wattmeter 
reading  is  zero,  and  the  power 
in  the  circuit  is  equal  to  the 
reading  of  the  second  instru- 
ment. If  the  lead  is  more  than 
60°,  the  reading  of  the  first  instrument  is  negative  and  of  the 


Fig.  241.  — Curves  showing  Relation  of 
Wattmeter  Readings  to  Lag  Angle  in 
Balanced  Three-phase  Circuit. 


398 


ALTERNATING  CURRENTS 


second  positive,  and  the  power  in  the  circuit  is  equal  to  the 
difference  of  the  two  readings.  When  the  angle  of  lag  is  ± 90°, 
the  readings  of  the  two  instruments  are  equal,  but  one  is  posi- 


to  Power  Factor  in  Balanced  Three-phase  Circuit. 

tive  and  the  other  negative.  The  relations  of  the  wattmeter 
reading's  to  the  angle  of  lag  between  + 90°  and  — 90°  are  shown 
by  the  curves  in  Fig.  241.  These  are  two  equal  sinusoids  with 
a phase  difference  equal  to  60°.  The  readings  of  two  watt- 


POLYPHASE  CIRCUITS  AND  MEASUREMENT  POWER  399 


meters  in  a balanced  three-phase  circuit  at  any  value  of  the  lag 
are  in  the  proportion  of  the  corresponding  ordinates  of  the  two 
curves.  Figure  242  shows  similar  relations  plotted  in  a different 
manner.  In  this  case  the  ratios  of  the  readings  of  the  watt- 
meters are  plotted  as  ordinates  and  the  power  factors  as 
abscissas.  This  curve  is  derived  directly  from  the  curves  of 
Fig.  241. 

The  same  conditions  obtain  in  an  unbalanced  circuit,  pro- 
vided equivalent  angles  of  lag  are  considered. 


CHAPTER  IX 


HYSTERESIS  AND  EDDY  CURRENT  LOSSES 

106.  The  Energy  Losses  Caused  by  Magnetic  Hysteresis.  — 

When  the  number  of  lines  of  force  passing  through  a coil  is 
changed,  a voltage  is  generated  in  the  coil  the  value  of  which  is 

e = — in  which  is  the  change  per  second  of  the 

10s  dt  dt  b L 

number  of  magnetic  linkages  through  the  turns  of  the  coil. 

If  a current  is  flowing  in  the  coil  when  the  change  occurs,  the 

work  dw , done  upon  or  by  the  lines  of  force,  is  equal  to  eidt, 

and  therefore,  dw=  — But  in  case  the  magnetic  circuit 

is  of  uniform  material  and  its  cross  section  is  of  uniform  area 

d>  = AB  and  d<b  = AdB,  and  n=  , where  <£  is  the  mag- 

4l7TI 

netic  flux  in  lines  of  force,  B is  the  magnetic  density  (lines  of 
force  per  square  centimeter),  A is  the  cross  section  of  the  mag- 
netic circuit  in  square  centimeters,  M is  magnetomotive  force 

in  Gilberts  (one  Gilbert  = ampere-turns),  and  n is  the  num- 

ber  of  turns  in  the  coil.  From  this  we  have  dw  = iedt  = 

— AMdB  j$ut  in  any  space  sufficiently  limited  so  that  H, 

107  x 4 ir 

the  field  strength  or  magnetizing  force,  is  uniform,  M = H L, 
where  L is  the  length  of  the  magnetic  circuit,  and  conse- 

ciuentlv  dw  = ~ HdB.  It  will  be  observed  that  AL  is 

1 J 107  x 477- 

equal  to  the  volume  of  the  magnetic  circuit  under  considera- 
tion, and  the  coefficient  of  HdB  is  therefore  proportional  to 
that  volume.  By  integrating  this  we  get  the  expression 

W=  f iedt  = - *L—  f HdB. 

J 107  X 4 7T*' 


400 


HYSTERESIS  AND  EDDY  CURRENT  LOSSES 


401 


The  power  in  watts  which  is  expended  during  the  change  is 
of  course  the  average  rate  of  doing  the  work  shown  above  and 
is  equal  to  the  work  divided  by  t.  This  work  is  taken  from  the 
electric  cii’cuit  as  the  magnetism  rises  and  returned  to  the  cir- 
cuit as  the  magnetism  falls,  when  the  varying  magnetism  is 


f 


HdB 


B 

r 

s 

/ 

— -/<J 

0 

H 

Fig.  243.  — Curve  of  Rising  and 
Falling  Magnetism. 


caused  by  varying  the  exciting  current.  The  value  of 
equals  an  area  between  curves  of  rising  and  falling  magnetism, 
such  as  Ogjs  of  Fig.  243,  when  the  operation  starts  with  both 
H and  B of  zero  value,  proceeds  to  y 

a value  of  H equal  to  rj  and  the 
corresponding  value  of  B equal  to 
Or , and  returns  to  the  condition  of 
H equal  to  zero  and  the  state  of 
remanent  magnetism  (as  might  be 
the  case  if  the  magnetic  circuit 
was  composed  of  iron)  when  B 
equals  Os.  This  work  is  stored 
in  the  magnetic  linkages  or  ex- 
pended in  heating  the  metal.  That  some  of  the  work  remains 
stored  in  the  magnetic  linkages  under  the  conditions  indicated 
by  Fig.  243  is  manifest  by  the  finite  value  Os  of  the  magnetism 
at  the  end  of  the  procedure ; and  part  of  this  stored  energy  is 
recoverable,  but  it  is  impossible  to  say  from  the  given  premises 

what  proportion  of  the  whole  it  con- 
stitutes. However,  when  the  j HdB 

is  carried  through  a closed  (cyclic) 
curve,  such  as  is  illustrated  in  Fig. 
244  for  instance,  the  available  energy 
of  the  magnetic  linkages  at  the  end  of 
the  cycle  is  the  same  as  at  the  begin- 
ning, and  in  that  case  all  of  the  energy 

represented  by  — --  J HdB  is  con- 

verted into  heat.  In  Fig.  244  the 
cyclic  curve  is  JSBjsbJ.  After  the 
iron  part  of  a magnetic  circuit  has 
come  into  a cycle  of  this  form  through  continuous  reversals 
of  the  magnetism  between  the  fixed  limits  of  J and  j,  the 

curve  repeats  itself  with  each  reversal,  and  j HdB  has  a fixed 

2d  J 


Fig.  244.  — Cyclic  Curve  of 
Magnetism. 


402 


ALTERNATING  CURRENTS 


value  per  cycle.  Upon  first  starting  the  cyclic  magnetization, 
B = / (-ff)  travels  over  a line  such  as  Ooj  of  Fig.  243,  but  it 
takes  up  the  fixed  cyclic  curve  when  the  iron  is  subjected  to 
complete  cycles  or  reversals  of  magnetism. 

A mental  picture  of  these  phenomena  may  be  formed  by  con- 
sidering that  work  is  done  upon  the  magnetic  molecules  in 
order  to  swing  them  into  line  when  a piece  of  iron  is  mag- 
netized. Part  of  this  work  is  expended  in  overcoming  an  oppo- 
sition to  the  swinging  of  the  molecules  which  is  akin  to  friction, 
and  this  work  is  converted  into  heat ; but  another  part  is  ex- 
pended in  overcoming  the  live  forces  of  mutual  attraction  and 
repulsion  between  the  molecules  or  groups  of  molecules,  and 
this  part  is  returned  to  the  electric  circuit  by  any  molecules 
which  swing  back  upon  the  withdrawal  of  the  magnetizing  force. 
As  no  energy  is  required  to  maintain  magnetic  flux  when 
once  established,  there  would  be  no  loss  of  energy  from  mag- 
netizing and  then  demagnetizing  a piece  of  iron  if  the  curves  of 
magnetism  ascending  and  descending  were  the  same,  for  in  this 

case  jHdB  for  the  descending  curve  would  be  equal  and 
opposite  to  CffdB  for  the  ascending  curve,  and  the  two 


would  cancel.  But  in  fact  there  is  a difference  between  the 
ascending  and  descending  branches  of  the  curves  of  magneti- 
zation of  iron  and  other  magnetic  metals,  which  is  caused  by 
Hysteresis  and  there  must  be  a loss  of  energy  due  to  the  oper- 
ation of  magnetizing  and  demagnetizing,  which  is  proportional 
to  the  area  between  the  ascending  and  descending  curves. 


In  Fig.  243  the  area  Orj  is  equal  to  j'HdB  going  up  the  curve 
and  srj  is  equal  to  ^ lid B going  down  the  curve.  The  work 


which  was  expended  in  going  up  the  curve  to  magnetizing 
force  H and  magnetic  density  B is  proportional  to  area  Orj; 
and  the  work  which  was  recovered  in  going  down  the  curve 
to  zero  magnetizing  force  is  proportional  to  area  srj.  The 
net  work  which  has  been  expended  in  the  operation,  part 
of  which  may  yet  be  recoverable,  though  a portion  of  it 
has  been  expended  in  overcoming  what  may  be  called  mag- 
netic molecular  friction  and  is  not  recoverable,  is  propor- 
tional to  the  area  Ojs.  The  work  proportional  to  the  area  Ojs 


is  W = 


— AL 

107  X 4 7T 


The  recoverable  part  of  this  is 


HYSTERESIS  AND  EDDY  CURRENT  LOSSES 


403 


retained  by  tbe  remanent  (residual)  magnetic  flux  which  has 
a density  B = Os.  Of  this  part,  all  may  he  recovered  except 
that  which  is  absorbed  in  molecular  friction  when  the  coer- 
cive force  is  overcome  and  the  magnetism  is  forcibly  brought 
to  zero. 

If  the  piece  of  iron  is  carried  through  a cyclic  curve  of 
magnetization  so  that  it  coincidently  returns  to  the  same  values 
of  magnetization  and  magnetizing  force  as  those  from  which  it 
started,  all  the  work  which  has  been  expended  must  have  been 
done  at  the  expense  of  the  magnetising  circuit  and  is  irrecov- 
erable, as  already  explained.  Consequently,  in  a cyclic  curve 
like  Fig.  244,  the  area  of  the  cyclic  curve  is  directly  pro- 
portional to  the  lost  energy,  and  is  equal  to  the  lost  energy  in 
ergs  when  II  is  measured  in  gilberts  per  centimeter,  B is  given 
in  lines  of  force  per  square  centimeter,  L is  given  in  centi- 
meters, and  A is  given  in  square  centimeters.  If  the  cycles 
are  produced  by  reversals  of  the  magnetization  between  equal 
positive  and  negative  values  by  rotating  a cylindrical  piece  of 
iron  between  fixed  magnet  poles  or  subjecting  the  iron  to  the 
magnetizing  effect  of  a coil  carrying  alternating  currents,  the 
irrecoverable  energy  has  a fixed  value  per  cycle  as  shown  by 
the  formula,  which  is  proportional  to  the  area  of  the  cyclic 
curve  of  magnetization  or  “hysteresis  loop,”  and  this  area 
bears  a close  relation  to  the  maximum  magnetic  density  between 
the  limits  of  which  the  cycle  of  magnetism  is  carried. 

This  energy  loss  due  to  hysteresis  was  shown  by  Steinmetz, 
as  the  result  of  a very  brilliant  series  of  experiments,*  to  be 
closely  proportional  to  the  1.6  power  of  the  maximum  magnetic 
density,  within  the  ranges  of  density  used  in  ordinary  engineer- 
ing practice.'  Later  experiments  carried  on  by  Ewing  and 
others,  including  Steinmetz,  have  shown  that  the  exponent  need 
not  be  exactly  1.6,  but  commonly  approximates  thereto,  and  for 
most  practical  purposes  a figure  between  1.55  and  1.6  is  suf- 
ficiently close.  At  magnetic  densities  exceeding  20,000  lines 
of  force  per  square  centimeter,  the  exponent  seems  to  become 
considerably  smaller ; and  it  varies  for  different  steels. 

Warburg  called  early  attention  to  the  fact  that  the  actual 
loss  of  energy,  when  a metal  subject  to  hysteresis  is  carried 
through  a complete  magnetic  cycle,  is  proportional  to  the  area 

* Trans.  Amer.  Inst.,  Elect.  Eng.,  1892,  Yol.  IX. 


404 


ALTERNATING  CURRENTS 


of  the  cyclic  curve  of  magnetism.  This  was  also  demonstrated 
by  Ewing  independently  of  Warburg.* 

Combining  the  results  of  the  theoretical  considerations,  which 
show  that  the  area  of  a cyclic  curve  of  magnetization  is  pro- 
portional to  energy  per  unit  volume  of  the  iron  expended  on 
something  analogous  to  molecular  friction,  and  the  experi- 
mental considerations  which  show  that  the  area  is  proportional 
to  approximately  the  1.6  power  of  the  maximum  density  of  the 
magnetic  cycles,  gives  the  following  relations  for  the  hysteresis 
loss  per  cycle  in  a piece  of  iron: 

W — ClIdB  = vVBx  x 10-7, 

10'  X 47 rj+s 

in  which  rj  is  a numerical  quantity  depending  on  the  quality  of 
the  particular  sample  of  iron  and  is  often  called  the  “hysteresis 
constant  ” of  the  iron,  and  V is  the  volume  of  the  iron.  The 
numerical  value  of  g for  any  sample  of  iron  also  depends  upon 
the  unit  of  volume  adopted,  and  the  values  hereinafter  given 
assume  the  volumes  to  be  measured  in  cubic  centimeters. 

The  expression  is  ordinarily  written 

W=  gVB 16  x 10-7, 

since  the  exponent  x has  approximately  the  value  of  1.6  ; and  the 
hysteresis  loss  in  watts  is  equal  to  this  loss  per  cycle  multiplied 
by  the  number  of  cycles  per  second,  or 

P = vVfB 16  x 10-7, 

where  / is  the  frequency  of  the  magnetic  cycles. 

When  put  in  words,  this  is  sometimes  referred  to  as  Stein- 
metz’s  law;  namely,  the  hysteresis  loss  in  each  unit  volume  of 
iron  is  equal  to  the  product  of  (1)  a hysteresis  constant  fixed  by 
the  quality  of  the  iron  under  consideration , (2)  the  frequency  of 
the  magnetic  cycles . and  (3)  approximately  the  1.6  power  of  the 
maximum  magnetic  density  occurring  per  cycle. 

107.  Curves  of  Hysteresis,  and  of  Current  and  Voltage  dis- 
torted by  Hysteresis. — Figure  245  illustrates  cyclic  hysteresis 
curves  for  the  same  test  piece  with  varying  degrees  of  satura- 


* Ewing,  Magnetic  Induction  in  Iron  and  Other  Metals,  p.  99. 


HYSTERESIS  AND  EDDY  CURRENT  LOSSES 


405 


tion,  when  produced  by  alternating  magnetizing  currents  with 
a wave  form  in  which  the  ordinates  increase  continuously  to 
a maximum  and  then 
similarly  decrease  to 
zero,  as  in  a sine  or  a 
parabolic  wave.  The 
three  cyclic  curves 
correspond  to  the  three 
exciting  currents  af- 
fording maximum  field 
strengths,  respectively, 
of  two,  five,  and  fifteen 
gilberts.  , 

Figure  246  shows  a 
similar  curve  resulting 
from  a magnetizing 
wave  which  contains 
marked  indentations, 
such  as  the  current 
illustrated  by  curve  I 
in  Fig.  247 ; that  is, 
the  wave  does  not  in- 
crease continuously  to 
a maximum,  but  has 
ripples  superposed  on  its  form.  It  will  be  noted  here  that  the 
cyclic  hysteresis  curve  contains  small  auxiliary  loops.  The 
area  now  is  proportional  to  the  area  of  the  main  cyclic  curve 

plus  the  sum  of  the  areas  of  the 
auxiliary  loops.  Figure  247  shows 
the  current  wave  used  in  the  last 
series  resolved  into  its  harmonics, 
which  consist  of  a fundamental  and 
a fifth  harmonic  of  considerable 
amplitude. 

If  an  air  space  exists  in  series 
with  the  iron  part  of  a magnetic 
circuit,  its  effect  is  to  increase  the 
exciting  current  which  is  required 
to  produce  a given  maximum  mag- 
netic density ; but,  except  so  far  as 


Fig.  240. — Hysteresis  Curve 
caused  by  a Wave  of  Mag- 
netizing Current  which  has 
more  than  One  Maximum. 


Fig.  245.  — Hysteresis  Curves  corresponding  to 
Three  Maximum  Magnetic  Densities,  each  pro- 
duced by  an  Alternating  Magnetizing  Current 
with  a Single  Maximum  Value. 


406 


ALTERNATING  CURRENTS 


there  is  a greater  loss  in  the  electric  circuit  due  to  the  larger 
current,  it  does  not  increase  the  active  component  of  the  excit- 

ing  current,  inasmuch  as  the  area 
of  the  hysteresis  cycle  is  not 
altered,  assuming  that  the  iron  is 
not  changed  in  volume  or  con- 
formation and  the  same  maximum 
magnetic  density  is  maintained. 
This  case  may  be  illustrated  by 
the  tilted  magnetic  curve  shown 
in  Fig.  248.  The  effect  of  the  air 
space  is  here  represented  by  the 
position  of  the  tilted  axis  OYt 
with  respect  to  the  axis  OY. 
Curve  bOb'  is  the  hysteresis  loop  of  the  ir-on  in  the  magnetic 
circuit,  and  the  abscissa  corresponding  to  any  ordinate  repre- 
sents the  magnetic  force  required  to  set  up  in  the  iron  the 
magnetic  density  corresponding  to  the  particular  ordinate. 
The  abscissa  of  the  line  GY1 
corresponding  to  the  same 
ordinate  represents  the  mag- 
netic force  required  to  set 
up, the  same  magnetic  den- 
sity in  the  air  space. 

Consequently  these  two  ab- 
scissas added  together  give 
the  magnetic  force  required 
to  set  up  that  magnetic  den- 
sity in  iron  and  air  space  in 
series.  It  is  thus  to  be  seen 
that  the  hysteresis  loop  for 
the  magnetic  circuit  com- 
posed of  iron  and  air  gap  in 
series  consists  of  a curve 
like  bx  0 b'v  which  has  abscissas  equal  to  the  sums  of  the  abscissas 
of  the  hysteresis  loop  of  the  iron  in  the  magnetic  circuit  and 
the  straight  line  which  corresponds  to  the  curve  of  magnetiza- 
tion of  the  air  gap.  Manifestly,  the  area  of  the  hysteresis  loop 
and  the  corresponding  hysteresis  loss  is  unaltered  by  the  intro- 
duction of  the  air  gap  in  the  magnetic  circuit,  provided  the 


Fig.  248.  — Hysteresis  Cycle  of  a Magnetic 
Circuit  containing  an  Air  Space. 


Fig.  247. — Component  Harmonics 
of  the  Magnetizing  Current  Giv- 
ing Hysteresis  Loop  of  Fig.  246. 


HYSTERESIS  AND  EDDY  CURRENT  LOSSES 


407 


limits  of  magnetization  are  kept  the  same  ; but  the  plotted 
curve  is  skewed  off  to  the  right  by  the  effect  of  the  air  gap, 
and  the  exciting  current  required  to  produce  the  maximum 
magnetic  density  is  increased  on  account  of  the  reluctance  of 
the  air  gap. 

108.  Similarity  between  Magnetic  and  Dielectric  Hysteresis 
and  Friction.  — All  magnetic  materials  seem  to  be  subject  to  the 
phenomena  attendant  upon  a seeming  molecular  magnetic  fric- 
tion which  are  collectively  called  hysteresis.  As  already  ex- 
plained, the  area  of  a cyclic  hysteresis  curve  is  proportional  to 
the  energy  which  is  converted  by  molecular  action  into  heat 
when  magnetic  material  is  carried  through  cycles  of  magneti- 
zation. In  the  course  of  extended  experiments  already  referred 
to,  Steinmetz  demonstrated  that  the  area  of  the  cyclic  curve 
for  iron  is  approximately  proportional  to  the  1.6  power  of  the 
maximum  magnetic  density  within  ordinary  ranges  of  density. 
Similar  experiments  were  repeated  by  Professor  J.  A.  Ewing, 
and  he  therein  demonstrated  that  the  exponent  1.6  was  not 
necessarily  fixed,  but  that  1.6  approximately  represents  the  cor- 
rect value.  In  some  of  Ewing's  experiments  the  results  were 
more  nearly  represented  by  the  exponent  1.55.  He  also  showed 
that  it  may  be  much  larger  (perhaps  more  than  2.5)  at  very 
low  densities.  All  experiments,  however,  go  to  show  that  the 
area  of  the  cyclic  hysteresis  diagram  for  iron  is  very  nearly  pro- 
portional to  the  maximum  magnetic  density  affected  by  an  ex- 
ponent of  1.6  within  the  range  of  densities  in  usual  engineering 
practice. 

Some  also  suppose  that  dielectric  hysteresis  is  subject  to  a 
similar  exponential  law.  When  a condenser  is  subjected  to 
alternating  voltage,  energy  is  absorbed  in  the  dielectric,  and 
this  energy  is  proportional  (according  to  the  results  of  certain 
experimenters)  to  the  maximum  charge  of  the  condenser,  — that 
is,  to  the  maximum  electric  stress  or  density  of  electrostatic 
induction  in  the  dielectric,  — raised  to  the  1.6  power.  It  is 
probable  that  the  rotation  of  a dielectric  body  in  an  electrostatic 
field  would  result  in  conversion  of  energy  due  to  dielectric  hys- 
teresis in  a manner  analogous  to  the  conversion  of  energy  due 
to  magnetic  hysteresis  when  iron  is  rotated  in  a magnetic  field. 
It  may  also  be  expected  that  the  losses  for  equal  frequencies 
and  comparatively  small  electrical  densities  should  be  practi- 


408 


ALTERNATING  CURRENTS 


cally  similar  if  a given  volume  of  dielectric  was  rotated  in  an 
electrostatic  field,  and  if  it  was  subjected  to  an  equal  frequency 
of  equal  maximum  electric  stress  by  means  of  an  alternating 
electrostatic  field.  In  the  case  of  the  rotating  dielectric  the  in- 
crease of  loss  should  tend  to  become  less  and  less  as  the  electric 
stress  is  increased,  until  with  a certain  value  of  the  electric 
stress  the  hysteresis  loss  should  become  practically  zero.  This 
is  analogous  to  the  conditions  that  exist  with  respect  to  magnetic 
hysteresis,  and  of  course  can  only  be  true  in  case  the  molecules 
act  toward  each  other  in  a similar  manner. 

The  foregoing  conditions  also  have  analogies  in  mechanical 
friction,  which  has  been  experimentally  proved  to  be  affected 
by  an  exponential  law.  Thus,  for  instance,  a fluid  friction,  as 
that  of  mercury  at  low  velocity,  has  been  experimentally  shown 
to  vary  approximately  in  proportion  to  the  1.6  power  of  the  pres- 
sure. This  points  the  way  to  an  inference  that  the  molecular 
actions  which  cause  absorption  and  conversion  of  energy  by  fric- 
tion may  prove  to  be  of  the  same  fundamental  nature  and  of  the 
same  order  as  the  molecular  actions  which  cause  the  absorption 
and  conversion  of  energy  through  magnetic,  and  perhaps  dielec- 
tric, hysteresis.  Professor  Ewing  has  suggested  that  experiments 
of  his  substantiate  this  view,  but  no  complete  demonstration  of 
these  relations  has  yet  been  disclosed.  Professor  Steinmetz, 
whose  accomplishments  in  advancing  alternating  current  phi- 
losophy have  been  varied  and  great,  has  cast  doubt  on  the  rela- 
tionship of  dielectric  hysteresis  to  the  other  phenomena,  and  some 
of  the  later  experiments  seem  to  show  that  what  was  thought  by 
some  to  be  dielectric  hysteresis  is  in  reality  only  the  effect  of  cur- 
rent seepage  in  the  dielectric  and  is  not  of  the  nature  of  hysteresis. 

109.  The  Magnetic  and  Dielectric  Hysteresis  Constants.  — It 
is  easy  to  compute  the  energy  which  is  absorbed  through  the 
effect  of  magnetic  or  dielectric  h}'steresis  and  converted  into 
heat  thereby,  provided  measurements  have  been  performed  on 
materials  similar  to  the  samples  under  consideration,  so  that 
the  coefficient  y giving  the  heat  developed  per  cycle  in  unit 
volume  of  the  material  at  unit  magnetic  density  is  known. 
This  coefficient,  as  already  pointed  out,  is  often  called  the  Hys- 
teresis constant  of  the  material. 

The  constants  of  dielectric  hysteresis  have  not  been  deter- 
mined satisfactorily.  It  may  be  assumed  that  the  dielectric 


HYSTERESIS  AND  EDDY  CURRENT  LOSSES 


409 


constant  varies  for  different  dielectric  materials,  but  the  evi- 
dence of  such  experience  as  has  been  had  in  the  commercial 
world  with  condensers  constructed  with  the  ordinary  dielectric 
substances  indicates  that  they  do  not  differ  very  widely  among 
themselves  in  respect  to  losses  per  cycle,  unit  volume,  and  unit 
electrostatic  stress  that  may  be  due  to  hysteresis  effects. 

Large  numbers  of  experiments  have  been  made  and  are  being 
made  by  manufacturers  of  electrical  machinery  to  determine  the 
constants  of  magnetic  hysteresis  related  to  various  qualities  of 
iron  and  steel.  Experiments  on  other  magnetic  materials  ex- 
cept iron  and  steel  have  been  few.  Cobalt  and  nickel  have 
been  tested  by  Professor  Ewing  and  others,  and  certain  other 
materials  have  been  tested;  but  none  of  these  have  any  impor- 
tant significance  in  theory  or  practice  at  present.  The  range 
of  hysteresis  constants  of  iron  is  well  represented  in  the  accom- 
panying table.  The  better  qualities  of  iron  are  such  as  would 
prove  satisfactory  for  use  in  alternating  current  transformers, 
alternating  current  motors,  and  the  like. 

The  following  table,  compiled  by  Messrs.  Swenson  and 
Erankenfield,  indicates  in  round  figures  the  values  of  the  coeffi- 
cient r)  for  various  irons ; but  it  is  to  be  remembered  that 
variation  in  the  processes  of  manufacture  or  in  the  treatment, 
as  well  as  in  the  composition  of  the  metal,  often  make  compara- 
tively large  changes  in  rj.  A small  addition  of  nickel  or  silicon 
to  soft  iron  intended  for  rolling  into  sheets  tends  to  reduce  the 
value  of  7]  and  increase  the  electrical  resistance  of  the  iron,  and 
such  silicon  iron  is  now  generally  used  for  the  cores  of  trans- 
formers and  the  like. 

f 

TABLE  OF  HYSTERESIS  CONSTANTS 


Best  Wrought  Iron  and  Steel  Sheets 001 

Good  Soft  Iron  and  Steel  Sheets 002 

Ordinary  Soft  Iron 003 

Annealed  Cast  Steel  008 

Ordinary  Cast  Steel .........  .012 

Ordinary  Cast  Iron  .........  .016 

Hard  Cast  Iron  and  Tempered  Cast  Steel  ....  .025 


The  constants  in  this  table  are  the  values  of  rj  to  be  used  in 
the  formula  with  cycles  per  second,  volume  in  cubic  centimeters, 
and  magnetic  density  in  lines  of  force  per  square  centimeter. 


410 


ALTERNATING  CURRENTS 


110.  Measurement  of  the  Energy  absorbed  by  Hysteresis. — 

Methods  of  obtaining  the  approximate  values  of  magnetic 
hysteresis  constants  of  samples  of  iron  are  now  so  simple  that 
every  well-equipped  electrical  engineering  laboratory  is  fitted 
for  such  work,  and  a large  proportion  of  the  manufacturers 
carry  out  systematic  tests  upon  their  iron.  Some  of  the  methods 
of  measurement  follow. 

a.  Some  of  the  manufacturers  do  not  make  tests  with  special 
instruments,  but  punch  a certain  volume  of  iron  sheets  into  trans- 
former stampings  and  determine  the  hysteresis  loss  by  a watt- 
meter when  the  magnetic  density  is  at  a fixed  value.  For  this 

purpose  the  stampings  are  built 
up  into  a closed  magnetic  circuit 
of  constant  cross  section.  There 
is  some  uncertainty  in  determin- 
ing the  value  of  the  magnetic 


density,  since  this  depends  upon 
the  indications  of  a voltmeter  and 
is  affected  by  the  form  of  the  im- 
pressed voltage,  because  the  maxi- 
mum magnetic  density  depends 
upon  the  area  of  the  voltage  wave 
instead  of  upon  its  effective  value 
which  is  indicated  by  the  volt- 
meter. If  two  samples  of  iron 
are  tested  by  this  means,  using  the 
same  exciting  voltage,  the  results 
of  the  test  are  of  course  compara- 
tive ; but  exact  results  cannot  be 
obtained  readily  unless  the  voltage 
impressed  upon  the  exciting  winding  is  a strict  sinusoid. 
Figure  249  illustrates  the  arrangement  of  apparatus  for  this 
mode  of  testing.  The  method  is  much  used  iu  commercial 
testing  by  manufacturers  of  electrical  apparatus  who  only  care 
to  obtain  comparative  data  relating  to  the  iron  which  they 
use.  It  has  the  advantage  of  testing  the  iron  under  conditions 
quite  similar  to  those  under  which  it  is  used  iu  transformer 
cores  and  other  such  apparatus.  If  care  is  taken  to  impress  a 
sinusoidal  voltage  wave  on  the  coil  used  for  exciting  the  test 
sample,  the  method  can  be  used  with  reasonable  accuracy. 


Fig.  249.  — Diagram  of  Arrange- 
ments for  approximately  meas- 
uring Hysteresis  Losses. 


HYSTERESIS  AND  EDDY  CURRENT  LOSSES 


411 


The  hysteresis  constant  is 

P X 107 
71  ~ VfB1-6  ’ 

where  P is  the  power  shown  by  the  wattmeter  after  the  copper 
loss  in  the  exciting  coil  and  any  loss  from  eddy  currents  in 
the  stampings  have  been  deducted,  Pis  the  volume  of  the  iron 
in  cubic  centimeters,  / is  the  frequency,  and  B is  the  maximum 
magnetic  density  in  lines  of  force  per  square  centimeter.  The 
magnetic  density  is  obtained  from  the  formula 

rr_  irnBAf  # 

' 108 

where  n is  the  number  of  turns  of  the  exciting  coil,  A is  the 
cross  section  of  the  core  in  square  centimeters,  and  B is  the 
voltage  measured  by  the  voltmeter.  This  is  the  same  formula 
as  that  earlier  developed  in  the  discussion  of  alternator  voltage. 
The  symbol  B in  the  formula,  stands  for  the  induced  voltage  set 
up  in  the  exciting  coil  by  the  alternating  magnetism  excited  by 
alternating  current  with  a frequency  of  f periods  per  second 
flowing  in  the  coil.  This  is  substantially  equal  to  the  volt- 
meter reading,  if  the  IR  drop  in  the  resistance  of  the  coil  is 
negligible,  and  it  is  usual  to  assign  to  B the  voltage  read  on 
the  voltmeter  when  testing  according  to  this  method.  Caution 
must  be  used,  however,  to  avoid  appreciable  IR  drop  in  the 
coil  because  B in  the  formula,  strictly  speaking,  is  equal  to  the 
vector  difference  of  the  voltmeter  reading  and  the  IR  drop. 

The  foregoing  method  of  measurement  includes  the  eddy 
current  losses,  and  the  distribution  of  the  magnetism  through 
the  cross  section  of  the  test  sample  may  be  disturbed  by  the 
magnetic  effect  of  the  eddy  currents.  The  magnetism  set  up 
at  any  point  in  the  cross  section  is  dependent  on  the  resultant 
magnetic  force  at  that  point,  and  the  eddy-current  ampere  turns 
may  be  in  substantial  opposition  to  the  ampere  turns  of  the 
exciting  coil.  The  eddy  currents,  if  of  appreciable  magnitude, 
therefore  tend  to  reduce  the  magnetic  density  in  the  interior  of 
the  stampings,  f Eddy  current  losses  vary  as  the  product  of  the 
square  of  the  frequency,  the  square  of  the  maximum  magnetic 
density,  and  approximately  as  the  reciprocal  of  the  electrical 


* Art.  11. 


t Art.  113. 


412 


ALTERNATING  CURRENTS 


resistance  of  the  stampings,  so  that  they  may  be  calculated 
and  subtracted  from  the  wattmeter  reading.  However,  as  the 
laminations  used  in  electrical  machinery  should  be  sufficiently 
thin  to  reduce  the  eddy  current  losses  to  a small  fraction  of  the 
total  Iron  loss,  which  comprises  the  hysteresis  and  eddy  current 
losses,  the  wattmeter  reading  is  often  substantially  equal  to  the 
hysteresis  loss  alone.  If  the  frequency  of  the  alternating  cur- 
rent is  very  low,  the  eddy  currents  are  almost  always  negligible. 

Plans  have  been  proposed  for  separating  the  hysteresis  and 
eddy  current  losses  by  computations  based  on  test  data.  In 
one  process,  wattmeter  measurements  are  made  of  the  losses  at 
two  different  voltages.  The  magnetic  density  is  substantially 
proportional  to  the  impressed  voltage,  since  the  induced  voltage 
is  equal  to  the  vector  difference  of  impressed  voltage  and  IR 
drop  in  the  coil,  the  latter  being  of  small  value  by  the  con- 
struction of  the  apparatus;  and  the  other  factors  of  hysteresis 
and  eddy  current  losses  are  independent  of  the  voltage.  The 
following  expressions  therefore  connect  the  two  wattmeter  read- 

in°s:  P = aEl-6  + bE\ 

Px  = aE™  + bE !2, 

where  the  first  term  in  each  right-hand  member  represents  the 
hysteresis  loss  and  the  second  term  the  eddy  current  loss.  That 
E and  El  vary  as  the  magnetic  density  is  shown  by  the  voltage 
formula  given  earlier  in  the  article.  Hysteresis  loss  varies  as 
the  1.6  power  of  the  maximum  magnetic  density,  and  hence  as 
EIS  ; and  eddy  current  loss  varies  as  the  square  of  the  maximum 
magnetic  density,  and  hence  as  E2.  The  symbols  a and  b are 
quantities  depending  upon  the  characteristics  of  the  magnetic 
circuit  and  the  frequency,  and  are  assumed  to  be  constant. 
From  these  equations 

PE2~P,E2 
a E™Ef  - E2E™' 

P,  pi-6  _ PE,™ 

■ E2E2  6 — E'2E11S‘ 

As  P,  Px  and  P,  Ex  are  known  by  the  instrument  readings, 
the  values  of  a and  b may  be  computed.  The  hysteresis  losses 
for  the  two  voltages,  then,  are 

H = P - bE 2 and  Hx  = P1  - bE2 ; 


HYSTERESIS  AND  EDDY  CURRENT  LOSSES 


413 


and  the  eddy  current  loss  for  each  voltage  is 

F=  P — aE1-6  and  Fx  = P1  — aE p1,6. 


Since  hysteresis  losses  are  proportional  to  the  1.6  power  of 
voltage  and  eddy  current  losses  to  the  square  of  the  voltage, 
the  computed  values  of  H,  Hv  F , and  Fl  ought  to  yield  the 
results : 


H = H}  Is  =Fi 

E i-«  E™  and  E2  E2' 


hut  the  fact  is  that  results  of  the  measurements  and  computa- 
tions will  seldom  bear  this  test.  The  errors  of  observation 
inherent  in  the  measurements  are  large  compared  with  the 
quantities  computed,  and  the  effects  of  the  errors  are  magnified 
by  tbe  fact  that  observed  quantities  come  into  the  computed 
results  by  differences.  The  consequence  is  that  this  process 
is  not  reliable  and  is  little  used. 

If  the  frequency  can  be  varied  without  altering  the  sine 
form  or  amplitude  of  the  voltage  wave,  the  iron  loss  for  different 
frequencies  f and  fx  may  be  written, 

P =cf-.G  + d, 

Pi = <?/r-6 + & 


where  the  first  term  in  each  right-hand  member  is  hysteresis  loss 
and  the  second  is  eddy  current  loss.  The  hysteresis  loss  varies 
proportionally  to/--6  because  it  is  proportional  to  the  product 
of  the  number  of  cjmles  per  second  and  _B1,6 ; but  B varies 
inversely  as  the  frequency  when  the  voltage  is  maintained 
constant  in  the  arrangement  of  apparatus  under  consideration, 
as  can  be  seen  from  the  formula  for  voltage  given  earlier  in 
the  Article.  The  part  of  the  iron  loss  due  to  eddy  currents 
should  remain  constant  under  the  conditions  here  considered, 
because  the  eddy  current  loss  is  directly  proportional  to  the 
product  of  frequency  squared  with  magnetic  density  squared, 
and  these  are  inversely  proportional  to  each  other  in  the 
arrangement  of  apparatus  under  consideration.  The  constants 
c and  d of  the  equations  depend  upon  the  characteristics  of  the 
magnetic  circuit  and  the  voltage.  Having  observed  P , Px  and 
/,  fv  c and  d may  be  computed  and  the  values  of  the  hysteresis 
loss  and  eddy  current  loss  separated  from  each  other  by  a 


414 


ALTERNATING  CURRENTS 


process  similar  to  that  indicated  on  the  preceding  page ; 
but  this  process  is  also  subject  to  the  causes  of  inaccuracy 
accompanying  the  companion  process. 

b.  Ewing  Hysteresis  Tester. — In  this  instrument  a perma- 
nent horseshoe  magnet  A,  Fig.  250,  is  balanced  upon  knife- 

edges  and  has  a pointer  attached 
to  it  which  swings  over  the  scale 
B.  The  test  piece,  made  up  of 
laminated  iron,  is  rotated  between 
the  pole  pieces  of  the  magnet.  If 
no  losses  were  present,  the  ten- 
dency for  the  pointer  to  swing  in 
one  direction  would  be  as  great  as 
in  the  other ; but  with  hysteresis 
present,  the  torque  upon  the  mag- 
net is  greater  when  the  test  piece 
is  leaving  the  magnet  than  when 
approaching,  and  the  average 
torque  exerted  on  the  magnet  is 
proportional  to  the  work  done  on 
the  test  piece  per  revolution. 
Since  the  hysteresis  loss,  that  is, 
the  poAver  absorbed  by  the  effect 
of  hysteresis,  is  directly  propor- 
tional to  the  speed,  the  torque  is  independent  of  the  speed  pro- 
vided disturbing  influences  like  eddy  current  losses  or  wind 
friction  are  not  encountered.  If,  then,  the  test  piece  is  rotated 
rapidly  enough  to  give  a steady  deflection  of  the  pointer  on  the 
magnet,  but  not  to  set  up  appreciable  eddy  currents  or  cause 
Avind  friction,  the  deflection  of  the  pointer  is  proportional  to 
the  hysteresis  loss. 

Test  pieces  of  known  hysteresis  constants  are  furnished  Avith 
the  instrument.  The  constants  of  unknown  samples  of  the 
same  length  and  approximately  the  same  cross  section  as  the 
standard  samples  can  be  determined  by  comparison  with  them. 
As  the  hysteresis  constant  is  approximately  the  same  for  all 
values  of  the  magnetic  density  within  the  range  of  ordinary 
commercial  practice,  the  use  of  a permanent  magnet  in  such  an 
instrument  seems  justified,  but  the  results  obtained  by  the  in- 
strument are  not  very  satisfactory.  This  is  doubtless  partially 


HYSTERESIS  AND  EDDY  CURRENT  LOSSES  415 

due  to  the  small  size  of  the  samples  that  may  be  used  for 
testing. 

c.  The  Holden  Tester . — In  the  Holden  hysteresis  tester  the 
pole  pieces  of  an  electro-magnet  A revolve  about  a test  piece 
B , built  up  from  disc  or  ring-shaped  stampings  as  shown  in  Fig. 
251.  The  magnetic  flux  traversing  the  test  piece  is  measured 
by  means  of  a coil  which  surrounds  the  core  and  is  cut  by  the 
flux  as  it  revolves.  This  coil  is  attached  through  a commutator 
to  a voltmeter.  The  density  of  the  magnetism  traversing  the 
test  piece  may  be  deter- 
mined from  the  observa- 
tions by  means  of  the 
formula 

B = ciTFH > 


where  E is  the  voltmeter 
reading,  S the  number  of 
test  coil  conductors  on 
the  surface  of  the  test 
piece,  V the  revolutions 
of  the  field  magnet  per 
minute,  and  A the  mean 
cross  section  of  the  test 
piece  in  square  centimeters.  The  test  ring  is  so  supported  that 
it  can  twist  against  a spring,  the  torsion  of  which  can  be  meas- 
ured. The  torque  acting  between  the  test  piece  and  the  field 
magnet  is  proportional  to  the  loss  due  to  hysteresis  and  eddy 
currents,  as  in  the  case  of  the  Ewing  tester,  and  can  be  deter- 
mined from  the  torsion  of  the  spring.  This  instrument  can 
be  used  either  for  comparison  of  different  samples,  by  observ- 
ing the  deflections  caused  by  each,  other  things  remaining 
equal,  or  for  the  direct  determination  of  hysteresis  constants  by 
computing  the  power  absorbed  by  the  hysteresis  per  cycle  and 
unit  of  volume  at  any  desired  magnetic  density.  The  apparatus 
has  the  advantage  of  permitting  frequency  and  magnetic  density 
to  be  varied  at  the  will  of  the  observer.  The  former  is  varied 
by  altering  the  speed  at  which  the  field  magnet  is  revolved  by  an 
outside  motive  power,  and  the  latter  is  varied  by  changing  the 


REVOLVING 


Fig.  251.  — Diagram  of  Holden  Hysteresis 
Tester. 


* Art.  11. 


416 


ALTERNATING  CURRENTS 


exciting  current  fed  to  the  field  magnet.  The  hysteresis  and 
eddy  current  losses  may  be  separated  at  constant  speed  by  ob- 
taining the  aggregate  loss  at  two  magnetic  densities,  the  former 
varying  as  B16  and  the  latter  as  B 2 ; or  at  constant  magnetic 
density  by  obtaining  the  aggregate  loss  at  two  speeds,  in  which 
case  the  hysteresis  loss  varies  as  V and  the  eddy  current  loss  as 
V2.  The  separation  of  the  two  losses  may  thus  be  established 
by  computations  similar  to  those  already  explained. 

d.  Step  by  Step  Method.  In  this  method  a core  A,  Fig. 
252,  is  inserted  in  a magnetizing  coil  B and  a measuring  coil 

C.  A ballistic  galvanom- 
eter B is  attached  to  the 
coil  C.  After  the  residual 
magnetism  of  the  sample 
has  been  destroyed  by 
some  process,  such  as  send- 
ing an  alternating  current 
through  the  exciting  coil 
and  gradually  reducing  it 
to  zero,  direct  current  is 
sent  through  the  coil  B, 
and  increased  by  conven- 
ient increments  until  the 
highest  desired  excitation 
is  reached.  The  exciting 
current  is  then  reduced  by  the  same  steps.  Upon  reaching 
zero,  circuit  connections  are  reversed  and  the  process  repeated. 
Each  change  of  the  exciting  current  causes  a change  in  the 
magnetism  set  up  in  the  test  piece,  which  in  turn  induces  volt- 
age in  the  test  coil  and  causes  a throw  of  the  galvanometer 
needle,  which  must  be  read  and  recorded  with  the  correspond- 
ing current  value.  The  magnetic  density  set  up  in  the  test 
piece  by  the  first  increment  of  current  may  be  computed  from 

the  relation  B = where  R is  the  resistance  of  the  galva- 

2 Anl 

nometer  circuit,  K the  galvanometer  constant,  /3  its  throw,  A the 
cross  section  of  the  test  piece,  and  nx  the  number  of  turns  on 
coil  C.  The  added  magnetic  density  for  each  additional  incre- 
ment of  current  may  be  computed  from  the  same  relation,  using 
the  corresponding  galvanometer  reading.  Plotting  the  mag- 


Fig.  252.  Step  by  Step  Method  of  Testing  for 
Hysteresis  Loss. 


HYSTERESIS  AND  EDDY  CURRENT  LOSSES 


417 


netic  densities  as  ordinates  with  currents  or  magnetic  force 


(/T  = _ 7rn^- , wpere  n is  tiie  number  of  turns  on  the  coil  B,  I is 


10* 


the  current  shown  by  the  amperemeter,  and  l is  the  average 
length  of  the  magnetic  circuit)  as  abscissas,  gives  the  hysteresis 
loop,  and  the  hysteresis  loss  per  cycle  may  be  estimated  from 
the  area  of  the  loop. 

It  is  desirable  to  repeat  the  step  by  step  current  cycle  a half 
dozen  or  more  times  before  taking  readings,  as  otherwise  the 
curve  may  not  be  either  closed  or  symmetrical  above  and  below 
the  axis  on  account  of  the  fact  that  the  first  cycle  of  magnetiza- 
tion starts  at  zero  //and  B and  is  therefore  unsymmetrical.  As 
the  reversals  proceed,  the  curve  gradually  assumes  the  form  of 
one  of  the  curves  shown  in  Fig.  245.  This  method  is  quite 
laborious  and  liable  to  inaccuracies  from  various  sources. 


111.  Computations  of  Eddy  Current  Losses.  — In  all  conduct- 
ing matter  placed  in  a variable  magnetic  field  there  are  in- 
duced, as  already  frequently  referred  to,  certain  Eddy  currents, 
or  Foucault  currents  as  they  are  often  called.  The  exact  values 
of  such  currents  cannot,  as  a rule,  be  computed,  because  the 
magnetic  densities  at  different  parts  of  a conductor  and  the  rates 
of  change  of  magnetism  cannot  be  fully  known,  nor  can  the 
exact  resistances  of  the  paths  of  flow  be  fully  determined. 
There  are  some  conditions  under  which  the  computation  of 
these  eddy  current  losses  is  of  value,  as,  for  instance,  in  a trans- 
former core  where  the  character  of  the  metal  used  is  well 
known  and  the  paths  of  the  magnetic  lines  of  force  can  be 
approximately  predicted.  Suppose  the  condition  of  a thin  plate 
of  iron  in  which  magnetic  lines  of  force  are  set  up  parallel  to 
the  sides  of  the  plate  and  the  magnetic  density  is  uniform  over 
the  cross  section.  When  this  magnetic  density  passes  through 
an  alteration  in  strength,  currents  tend  to  circulate  in  paths 
which  are  parallel  with  the  edges  of  the  cross  section  and  per- 
pendicular to  the  lines  of  force.  The  average  length  of  these 
paths  and  their  aggregate  cross  section  may  be  estimated  if  the 
dimensions  of  the  plate  are  known.  The  voltage  set  up  in 
each  portion  of  the  paths  may  also  be  calculated  if  the  maximum 
magnetic  density  and  its  rate  of  change  are  known.  If  we  are 
acquainted  with  the  specific  electrical  resistance  of  the  plate, 
we  can  also  determine,  with  a considerable  degree  of  approxi 

2 E 


418 


ALTERNATING  CURRENTS 


mation,  the  average  resistances  in  the  paths  of  the  eddy  cur- 
rents; and  we  are  therefore  enabled  to  compute  with  reasonable 
accuracy  the  amount  of  power  expended  on  account  of  the  flow 
of  these  currents.  We  can  do  the  same  when  the  magnetic 
core  is  composed  of  an  aggregation  of  cylindrical  wires.  In 
the  two  following  discussions  it  is  assumed  that  the  magnetic 
material  is  so  finely  subdivided  that  the  magnetic  flux  may  be 
considered  constant  over  the  cross  section  of  a lamination. 
This  is  permissible  in  view  of  the  practice  in  modern  manufac- 
ture of  electrical  machinery.  With  such  fine  subdivisions  of 
the  magnetic  circuit,  it  is  ordinarily  sufficiently  accurate  to 
assume  that  B = yH,  where  H is  the  impressed  field  strength 
and  y is  the  permeability  of  the  metal,  or  that  the  magnetic 
effect  of  the  eddy  currents  is  negligible. 

A.  Eddy  Current  Loss  in  Cylindrical  Conductors.  — Eddy  cur- 
rent loss  varies  directly  as  the  square  of  the  frequency,  directly 

as  the  square  of  the  maximum  mag- 
netic density,  and,  if  the  resistance  of 
the  circuits  in  which  the  eddy  currents 
flow  is  large  compared  with  their  react- 
ance, the  loss  varies  nearly  inversely 
with  the  resistance.  These  relations 
are  manifest  from  the  fact  that 


Fig.  253.  — Illustration  of 
Eddy  Currents  in  a 
Cylindrical  Conductor. 


P = I2R  = 


E2R 


E 2 


B + 


X 2’ 

It 


and  E is  proportional  to  the  rate  of 
change  of  the  magnetic  linkages  in 
the  circuits.  This  is  equally  true  whether  the  magnetic  core 
is  rotated  in  a fixed  magnetic  field  or  is  influenced  by  the  mag- 
netism induced  by  a coil  carrying  alternating  currents.  The 
resistance  of  the  eddy  current  circuits  is  dependent  upon  the 
specific  resistance  of  the  metal  concerned  and  the  volume  and 
conformation  of  the  metal  within  which  the  eddy  currents  may 
be  induced. 

Assume  that  the  large  circle,  Fig.  253,  represents  a cross 
section  taken  perpendicular  to  the  axis  of  a cylindrical  wire  of 
magnetic  metal  and  that  the  magnetism  passes  along  the  wire 
perpendicular  to  the  plane  of  the  paper  and  of  uniform  mag- 


HYSTERESIS  AND  EDDY  CURRENT  LOSSES 


419 


netic  density  over  the  cross  section.  Now  as  the  magnetism 
varies,  cylindrical  currents  will  be  set  up  which  are  projected 
in  Fig.  253  as  concentric  circles  in  the  cross  section  of  the 
wire.  If  we  consider  the  path  of  one  of  these  currents  of 
radius  r,  the  instantaneous  voltage  acting  in  the  circumference 
having1  this  radius  is 

0 dB  1n_a 
e = 7rr2  — - x 10  8 ; 
dt 

which,  when  the  magnetism  varies  as  a sine  function,  becomes 

e = t rr2  d(Bms[nn)  10-8  = o 7 r‘lrJBm  cos  a X 10"8, 
dt 


since  a = 2 7 rft;  and  the  effective  voltage  is 
AE=  V2  7 T2r2fBm  10-8. 


The  resistance  of  the  circular  paths  in  a unit  length  (centi- 
meter) of  an  elementary  cylinder  of  radius  r and  thickness  dr 

is  0 

R = 2 7 -rp 

1 1 x dr 


where  p is  the  specific  resistance  of  the  metal  in  ohms  per  one 
centimeter  of  length  and  one  square  centimeter  of  cross  section. 
The  instantaneous  power  (watts)  expended  in  a conductor  of 
this  resistance  due  to  a current  which  is  set  up  by  voltage 
having  the  value  given  in  the  first  of  the  foregoing  formulas, 
considering  the  resistance  of  the  eddy-current  paths  to  be  large 
compared  with  their  reactance,  is  approximately 

e 2 _ Trr3 (dB )2dr  10-16 

r~Ri~  -h*j2 

When  the  magnetism  is  sinusoidal,  the  average  power  (watts) 
expended  during  a period  in  the  filamentary  path  or  cylinder 
of  radius  r,  thickness  d>\  and  unit  length  (remembering  the 
assumption  that  the  magnetic  effect  of  the  eddy  currents  is 
negligible  in  comparison  with  the  inducing  magnetic  field),  is 

A 7> (AiT)2  _ 7r3r3(r?r)  f2BJ  1 0-16 

Rx 


P 


420 


ALTERNATING  CURRENTS 


and  the  joules  per  period  are 

A W=  AP  x T = 7r3r3^”> 2 1 ° ~ 16  dr ' 

P 

The  next  to  the  last  expression  gives  the  power  expended 
and  converted  into  heat  by  eddy  currents  in  an  elementary 
cylinder  of  unit  length,  thickness  dr , and  radius  r;  and  the  total 
power  expended  and  converted  into  heat  by  eddy  currents  in  the 
mass  of  the  solid  cylinder  of  conducting  material  of  radius  rl  and 
embraced  between  planes  one  centimeter  apart  is  equal  to  that 
expression  integrated  between  the  limits  of  0 and  rv  We 
therefore  find  that  the  loss  of  power  in  watts  which  is  caused 
by  eddy  currents  in  the  unit  length  of  the  wire  is 

p=  P^PBJPdr  _ 7r3PBmW 
Jo  10  Kp  4 p 1016  ’ 

and  the  work  in  joules  per  cycle  converted  into  heat  by  eddy 
currents  is 

W=  P T = 77-3 

4 p 1016  ’ 

T—  ~ being  the  length  of  one  period. 

The  strength  of  the  eddy  currents  may  be  found  in  a similar 
manner,  by  integrating 

j_  P' irfBmrdr  _ i rfBmr* 

A B1  Jo  V210 V 2 v/2  108  p 

To  find  the  watts  lost  per  cubic  centimeter  of  the  material  at 
frequency  f and  the  joules  lost  per  cubic  centimeter  of  the 
material  and  per  cycle  of  the  magnetism,  the  last  two  expres- 
sions but  one  may  be  divided  by  the  area  rrr\  of  the  end  of  the 
cylinder;  and  the  watts  expended  per  pound  of  material  at 
frequency  f,  or  the  joules  per  pound  of  material  and  per  mag- 
netic cycle,  may  then  he  obtained  by  multiplying  the  results  by 
the  number  of  cubic  centimeters  of  the  material  required  to 
make  a pound  in  weight. 

The  formulas  show  that  eddy  current  losses  in  a cylindrical 
wire  under  the  conditions  assumed  are  proportional  to  the 
fourth  power  of  the  diameter  of  the  wire.  If  a core  of  given 
configuration  is  to  be  made  up  of  equal  cylindrical  wires,  it  is 


HYSTERESIS  AND  EDDY  CURRENT  LOSSES 


421 


manifest  that  the  number  of  wires  composing  the  core  is 
substantially  proportional  to  the  reciprocal  of  their  diameter 
squared.  The  total  loss  in  a wire  core  of  given  configuration 
composed  of  equal  wires  is  therefore  proportional  to  the  square 
of  the  diameter  of  the  wires. 

It  is  to  be  noted  that  the  premises  used  in  the  analysis  include 
the  assumption  that  the  magnetic  lines  of  force  are  parallel  to 
the  walls  of  the  wires,  that  they  are  uniformly  distributed 
over  the  cross  section  of  the  wires,  that  they  vary  sinusoidally, 
and  that  the  magnetic  effects  of  the  eddy  currents  may  be 
neglected.  These  assumptions  cannot  be  accepted  as  rigidly 
holding  in  practice,  but  any  deviation  is  likely  to  occur  in  such 
a way  as  to  reduce  the  actual  losses  below  those  computed.  In 
particular,  the  magnetic  effects  of  the  eddy  currents  may  not  be 
negligible,  but  introduce  a disturbing  factor  which  reduces  the 

power  expended  from  to  ^ GGs  _ . This  effect  is  generally 

treated  as  though  the  eddy  currents  shielded  the  interior  of  the 
conductor  from  the  full  impressed  magnetic  force,  since  the 
magnetic  effects  of  the  eddy  currents  increase 
with  the  distance  from  the  surface  of  the 
conductor.* 

B.  Eddy  Current  Loss  in  Sheets  of  Rec- 
tangular Cross  Section.  — - A similar  integra- 
tion will  indicate  the  losses  in  the  case  of 
thin  sheets  of  rectangular  cross  sections  in 
which  the  lines  of  force  are  parallel  to  the 
sides  and  are  uniformly  distributed  over  the 
cross  section. 

In  a sheet  of  thickness  d let  a slice  be 
taken  a distance  x from  the  middle  and  of 
thickness  dx  as  indicated  in  Fig.  254,  which 
represents  the  sheet  with  its  edge  presented 
to  the  paper.  This  slice,  taken  in  connection 
with  a similar  one  on  the  other  side  of  the 
center  line,  may  be  considered  equivalent  to  a complete  ele- 
mentary electric  circuit  if  the  disk  is  thin  enough  so  that  the 
length  required  to  complete  the  circuit  at  the  ends  is  negligible. 


-d 
\/vvy 


'%d* 


i 

■I K 


Ax 

Fig.  254.  — Cross  Sec- 
tion of  a Disk  or 
Sheet  of  Iron  much 
magnified,  illustrat- 
ing Eddy  Current 
Circuits. 


* Art.  113. 


422 


ALTERNATING  CURRENTS 


The  instantaneous  number  of  lines  of  force  embraced  by  such  a 
circuit  is  2 IxB , where  l is  the  length  of  the  sheet  parallel  to  the 
page.  The  eddy  currents  flow  perpendicularly  to  the  lines  of 
force  and  parallel  to  the  sides  of  the  sheet,  up  on  one  side  and 
down  on  the  other  as  indicated  by  the  arrows  in  Fig.  254. 

The  instantaneous  voltage  set  up  by  varying  magnetism  in 
one  centimeter  of  the  elementary  slice  dx  is 

d -B  1 A —8 

e — x—  x 10  8- 
dt 


When  the  magnetism  varies  as  a sine  function,  this  is 

e = x s'“  u * x 10-8  = 2 7i fxBm  cos  « x 10-8, 
dt 

and  the  effective  voltage  is 

AJ2= 

108 

The  resistance  in  ohms  of  a portion  of  the  circuit  one  centi- 
meter long,  one  centimeter  deep,  and  dx  thick  is  lt1  = where 
I o is  the  specific  resistance  in  ohms. 

The  power  expended  in  the  elementary  slice  which  is  dx 
thick,  one  centimeter  long,  and  one  centimeter  deep  (again 
assuming  the  resistance  of  the  eddy  current  paths  to  be  large 
compared  with  their  reactance)  is 


A P = 


A E2  2 ir^PBJEdx 


* i 


1016  X P 


and  the  power  expended  in  a part  of  the  sheet  one  centimeter 
long,  one  centimeter  deep,  and  the  full  thickness  d is 


2 t rlPBJx^dx  = 2 7T2PBJd3 

J-id  10m p 4 x 8 x 1016  x p 


The  joules  per  cycle  converted  into  heat  by  the  eddy  cur- 
rents are 


w p -2^fB,:-dz 
f 4 x 3 X 1016  x P 


HYSTERESIS  AND  EDDY  CURRENT  LOSSES 


423 


The  strength  of  the  eddy  currents  may  be  found  in  a similar 
manner  by  integrating 

y A E _ C*d  V2  TrfB„,xdx  _ V2  7 rfBmd? 

. ~ ^ 7/  108  x}  ~ ~ 8 x 108  x p ' 

To  obtain  the  power  expended  per  cubic  centimeter  of  metal 
at  frequency/,  and  the  work  expended  per  cubic  centimeter  of 
metal  and  per  cycle  of  magnetism,  it  is  only  necessary  to  divide 
the  expressions  for  P and  W by  d.  This  gives  : 


where 


and 


7 T*PBJd2 

6 x 1016  x p 


S(fBmdy, 


6 x 10i6  x P ’ 


w 

Tf  cm 


fBJd 2 

6 x 1016  x p 


The  value  of  p for  soft  steel  and  iron  sheets  varies  from 
0.9  x 10~5  to  1.25  x 10~5  with  an  average  near  1 x 10-5 
ohms.* 

The  foregoing  computations,  like  those  relating  to  the  eddy 
current  losses  in  wires,  are  founded  on  certain  assumed  premises, 
including  the  assumptions  that  the  magnetic  lines  of  force  are 
parallel  with  the  sides  of  the  plate  and  uniformly  distributed 
over  its  cross  section,  and  that  the  magnetic  effects  of  the  eddy 
currents  are  negligible  in  comparison  with  the  impressed  mag- 
netic force.  Deviations  from  these  assumptions  in  commercial 
magnetic  cores  are  likely  to  cause  a reduction  of  the  actual 
losses  below  the  computed  values  rather  than  an  increase,  and 
the  formulas  are  therefore  safe  for  use  in  most  commercial 
instances.  The  magnetic  effects  of  the  eddy  currents  are  often 
of  considerable  importance  instead  of  being  negligible  and  are 
therefore  discussed  in  some  detail  in  Art.  113. 

The  difference  between  the  coefficients  of  r1  and  d in  the 
foregoing  formulas  for  the  eddy  current  losses  in  wires  and  in 

* Smithsonian  Physical  Tables,  3ded.,  p.  255;  Steimnetz,  Trans.  Amer.  Inst. 
Elect.  Eng.  Vol.  11,  p.  60d  ; London  Electrician,  Vol.  28,  p.  631  ; Loppe’  et 
Bonquet,  Courants  Alternatifs  Industriels,  p.  273. 


424 


ALTERNATING  CURRENTS 


rectangular  cross  sections  which  are  very  thin  compared  with 
their  length  shows  that  it  is  necessary  to  subdivide  a core  more 
finely  when  the  metal  is  in  plates  than  when  it  is  in  wires  if 
the  losses  are  to  be  equally  small,  but  the  greater  mechanical 
convenience  during  processes  of  manufacture  and  the  more 
satisfactory  mechanical  stability  which  may  be  obtained  by 
means  of  rectangular  plates  when  placed  in  actual  machinery 
has  led  to  an  almost  universal  adoption  of  plates  for  the  mag- 
netic cores  of  electrical  machines.  It  is  to  be  noted  that  the 
eddy  current  loss  in  a magnetic  core  of  fixed  dimensions  is 

proportional  to  the  square 
of  the  thickness  of  the 
stampings  used  to  build 
up  the  core. 

Figure  255,  following  a 
curve  calculated  by  Ewing, 
shows  the  relative  hys- 
teresis loss  and  eddy  cur- 
rent loss  in  lamince  of 
different  thicknesses  when 
the  maximum  magnetic 
density  is  4000  lines  of 
force  per  square  centimeter 
and  the  frequency  100 
periods  per  second.  The 
curve  which  starts  at  zero 
represents  the  eddy  current 
loss,  the  flatter  curve  repre- 
sents the  hysteresis  loss,  and  the  upper  curve  is  the  sum  of  the 
other  two  and  therefore  represents  the  total  iron  loss.  4 he 
curves  take  into  account  the  magnetic  effects  of  the  eddy 
currents,  which  become  important  when  the  sheets  exceed  a 
thickness  of  one  half  a millimeter.  The  hysteresis  curve  rises 
slightly  with  the  thickness  of  the  plates  on  account  of  crowding 
of  the  lines  of  force  by  the  magnetic  screening  effect  of  the 
eddy  currents.  The  figure  makes  it  plain  that  eddy  currents 
may  be  serious  sources  of  loss  in  the  magnetic  cores  of  electrical 
machinery,  and  that  it  is  desirable  that  the  iron  used  in  such 
cores  should  be  of  high  electrical  resistance  as  well  as  of  low 
hysteresis  constant. 


0 9 18  27  36  45  64  63  72  81 

THICKNESS  OF  PLATE  IN  MILS 

Fig.  255.  — Losses  due  to  Eddy  Currents  and 
Hysteresis  in  Sheet-iron  Plates,  produced 
by  Sinusoidal  Alternating  Magnetism  of 
100  Cycles  per  Second  and  4000  Lines  of 
Force,  Maximum  Density. 


HYSTERESIS  AND  EDDY  CURRENT  LOSSES 


425 


112.  Cyclic  Curves  of  Eddy  Currents.  — If  a periodic  mag- 
netization acts  upon  a laminated  iron  core,  the  eddy  current 
loss  produced  thereby  may  be  represented 
by  the  area  of  a loop  such  as  is  shown  in 
Fig.  256.  In  this  curve  the  ordinates  f\ 

represent  instantaneous  values  of  the  mag- 
netic density  B or  of  the  total  alternating 
flux  cj)  in  lines  of  force  inclosed  by  eddy 
current  circuits,  and  the  abscissas  repre- 
sent the  corresponding  values  of  induced  V/ 

eddy  currents  i in  amperes. 


If  e = ■ - is  the  voltage  causing  i to 

10  8dt  & 45 


flow,  it  is  easy  to  show  that  the  area  of  the 
cyclic  curve  of  Fig.  256  is  proportional  to 
the  power  expended  and  transformed  into 
heat  by  the  eddy  currents : The  instantaneous  value  of  this 


Fig.  256. — Loop  having 
an  Area  Proportional 
to  Eddy  Current  Loss 
occasioned  by  Sinu- 
soidal, Varying  Mag- 
netization. 


power  IS 


. . dd> 

v = ze  = i — , 
F 10  \it 


and  the  work  expended  during  a time  dt  is 

dW=iedt  = i4i- 
108 

The  differential  of  the  area  of  the  cyclic  loop  of  Fig.  256  is 

dA  = id(f>. 

Further,  the  work  expended  and  transformed  into  heat  by  the 
eddy  currents  during  one  cycle  of  the  magnetism  is 

A 

but  this  is  equal  to  . 

108 

If  the  abscissas  are  laid  off  on  the  scale  of  one  ampere  per  cen- 
timeter and  the  ordinates  on  the  scale  of  108  lines  of  force  per 
centimeter,  the  area  of  the  curve  in  square  centimeters  is  equal 
to  the  joules  of  eddy  current  loss  per  cycle.  If  other  scales 
are  used  in  plotting  the  curve,  the  area  must  be  multiplied  by 
a constant  in-order  to  arrive  at  the  joules  lost. 


426 


ALTERNATING  CURRENTS 


The  cyclic  curve  in  Fig.  256  is  of  the  form  that  results  when 
the  magnetism  alternates  as  a sine  function.  If  the  magnetic 
wave  is  irregular,  the  curve  may  lose  the  symmetrical  regularity 
notable  in  the  curve  of  Fig.  256,  since  the  abscissas  of  the  curve 
are  proportional  to  the  rate  of  change  of  the  magnetism.  Thus 
if  the  curve  of  magnetism  has  harmonics  which  cause  it  to 
have  two  maximum  points  per  half  cycle,  there  are  two  points  of 
rate  of  change  of  magnetism  and  current  i equal  to  zero,  which 
result  in  subsidiary  loops  at  the  top  and  bottom  of  the  cyclic 
curve.  Or,  if  the  maximum  value  of  the  curve  of  magnetism 
is  pushed  over  towards  « = 0°  or  « = 180°,  the  time  rate  of 
change  of  the  magnetism  is  decreased  in  one  part  of  the  period 
and  increased  in  another  part,  so  that  the  cyclic  eddy  current 
curve  is  skewed  with  reference  to  its  axis. 

An  exciting  coil  which  provides  the  impressed  magnetic 
force  in  a magnetic  circuit  must  carry  an  additional  energy 
component  of  current  on  account  of  the  eddy  current  loss  which 
is  approximately  „ 

where  P is  the  measured  or  calculated  value  of  the  eddy  cur- 
rent loss  in  watts,  and  Ex  is  the  induced  counter- voltage  in  the 
coil  (substantially  equal  and  opposite  to  the  impressed  line  volt- 
age in  a highly  inductive  circuit  as  in  the  case  of  the  unloaded 
primary  coil  of  a transformer).  The  combined  eddy  current 
circuit  is  similar  to  the  secondary  coil  of  a transformer  with  one 
turn  in  the  coil,  so  that  approximately 

I2=Ixn* 


where  I2  is  the  effective  value  of  the  current  in  the  combined 
eddy  current  circuit  and  n is  the  number  of  turns  of  wire 
in  the  exciting  coil  on  the  magnetic  core.  We  then  have 


approximately 


T>  Eo 

R,  = r 


A. 

n2Ix 


In  order  that  the  cyclic  curve  of  eddy  current  loss  may  be 
plotted,  it  is  necessary  that  the  instantaneous  values  of  mag- 
netism and  their  corresponding  rates  of  change  shall  be  known. 


* Art.  23. 


HYSTERESIS  AND  EDDY  CURRENT  LOSSES 


427 


B 


Fig.  257.  ■ 


■ Cyclic  Curve  of  Iron 
Loss. 


The  rates  of  change  are  directly  proportional  to  the  corre- 
sponding instantaneous  values  of  induced  voltage.  The  actual 
plotting  of  the  curve  is  not  usually  practiced,  but  the  relation- 
ships pointed  out  by  means  of  theo- 
retical considerations  are  valuable. 

If  the  abscissas  of  this  last  curve 
converted  into  terms  of  exciting 
current  are  added  to  the  corre- 
sponding abscissas  of  the  cyclic 
curve  of  hysteresis  for  any  par- 
ticular periodic  wave  of  magnetism, 
a cyclic  curve  is  obtained  like  that 
in  Fig.  257,  where  BTB'T'  is  the 
hysteresis  curve  and  BFB'F'  is  the 
combined  curve.  The  result  is  a 
loop  having  an  area  proportional  to 
the  total  iron  loss  in  the  circuit  and 
it  may  suitably  be  denominated  a 
Cyclic  curve  of  iron  loss. 

113.  Magnetic  Screening  due  to 
Eddy  Currents.  — The  law  of  the  magnetic  circuit  asserts  that 
the  magnetism  in  any  part  of  the  circuit  is  equal  to  the  net 
magnetomotive  force  in  that  part  of  the  circuit  divided  by  the 
magnetic  reluctance  thereof.  The  net  magnetomotive  force  is 
the  vector  sum  of  all  magnetic  forces  acting  on  the  circuit. 
When  a varying  magnetic  flux  is  set  up  in  material  which  is 
an  electrical  conductor,  the  filaments  of  varying  magnetism 
become  surrounded  by  eddy  currents,  and  the  magnetic  effects 
of  the  eddy  currents  may  be  quite  large  and  even  comparable  to 
the  impressed  magnetic  force  when  the  material  is  in  large 
masses  and  of  high  electrical  conductivity.  It  is  experi- 
mentally well  known  that  a copper  or  other  highly  conducting 
plate  placed  in  an  alternating  magnetic  field  has  eddy  currents 
set  up  in  it  to  so  high  a degree  that  their  magnetic  effect  may 
practically  neutralize  at  the  back  of  the  plate  the  magnetic  effect 
of  the  impressed  magnetic  field,  and  the  space  beyond  the  plate 
is  practically  screened  from  the  original  magnetic  field.  A 
non-magnetic  copper  plate  has  no  screening  effect  in  a magnetic 
field  that  is  constant,  and  its  screening  action  in  an  alternating 
magnetic  field  is  caused  by  the  magnetic  force  of  the  induced 


428 


ALTERNATING  CURRENTS 


eddy  currents.  This  is  in  accordance  with  the  theorem  that 
the  magnetic  force  of  induced  currents  must  tend  to  reduce  the 
impressed  magnetic  flux.  The  induced  voltage  causing  the 
flow  of  eddy  currents  is  in  lagging  quadrature  with  respect  to 
the  inducing  magnetic  field,  assuming  sinusoidal  functions ; 
and  if  the  path  of  the  eddy  currents  is  of  very  low  resistance, 
its  impedance  is  mostly  composed  of  the  reactance  component, 
and  the  eddy  currents  are  lagging  nearly  90°  behind  the  in- 
duced voltage.  Under  these  circumstances,  the  eddy  currents 
are  nearly  180°  in  phase  from  the  impressed  magnetic  force, 
and  their  magnetic  force  is  therefore  nearly  in  opposition  to 
the  impressed  force.  On  the  other  hand,  when  the  resistance 
of  the  eddy  current  circuit  is  high,  as  has  been  assumed  in  the 
computations  of  Art.  Ill,  the  currents  are  small  in  volume 
and  lag  little  behind  the  induced  voltage,  in  which  case  their 
magnetic  effect  is  small  and  nearly  in  quadrature  with  the 
impressed  magnetic  force.  It  then  has  little  effect  on  the 
magnetism  set  up. 

Certain  important  results  of  magnetic  screening  arise  in 
instances  where  the  magnetic  flux  is  constrained  within  a con- 
ductor in  a particular  manner,  as  in  the  case  of  the  magnetism 
in  the  laminated  iron  core  of  a transformer  or  a dynamo  arma- 
ture which  threads  through  the  stampings  parallel  to  their  flat 
sides.  A simple  example  is  a homogeneous  cylindrical  wire 
of  magnetic  metal  which  may  be  considered  as  a part  of  a 
magnetic  core  of  some  machine,  and  in  which  the  magnetism 
is  set  up  parallel  to  the  cylindrical  sides  of  the  wire  bv"  an  im- 
pressed magnetic  force  that  is  uniform  over  the  cross  section 
of  the  wire.  In  case  no  other  influences  came  into  play,  the 
magnetism  set  up  in  this  wire  would  be  uniform  over  its  cross 
section,  and  this  is  indeed  the  case  if  the  magnetizing  force  is 
constant ; but  if  the  magnetizing  force  is  alternating,  as  when 
the  exciting  current  is  alternating,  circular  eddy  currents  are 
set  up  in  the  conductor  which  themselves  produce  magnetic 
force  and  which  may  be  in  sufficient  volume  to  strongly  influ- 
ence the  net  magnetic  field  acting  on  the  iron  of  the  wire. 

Figure  258  shows  the  cross  section  of  such  a wire  in  which 
the  lines  of  force  are  supposed  to  be  perpendicular  to  the  plane  of 
the  cross  section.  The  magnetism  being  induced  by  an  alternat- 
ing exciting  current  is  itself  alternating  of  the  same  frequency, 


HYSTERESIS  AND  EDDY  CURRENT  LOSSES 


429 


and  induces  eddy  currents.  These  flow  in  circles  concentric 
with  the  surface  of  the  wire  as  illustrated  by  the  dotted  line 
in  Fig.  258,  and  their  volume  and  phase  relation  to  the  im- 
pressed magnetic  force  depend  on  the  impedance  of  the  eddy 
current  circuits.  The  magnetic  force  which  the  eddy  currents 
produce  within  the  mass  of  the  metal  is  parallel  to  the  axis  of 
the  wire,  and  its  magnitude  varies  along  a 

radius  from  a maximum  equal  to  v 2 ^ 77 ^ 

1 10 

gilberts  at  the  center  of  the  wire  (in  which 
I is  the  aggregate  volume  of  eddy  currents 
in  effective  amperes  per  centimeter  of  length 
of  the  wire)  to  a minimum  of  zero  at  the 
outer  surface  of  the  wire.  The  net  vector 
magnetic  force  at  any  point  in  the  cross 
section  is  equal  to  the  difference  between 
the  vector  impressed  magnetic  force  and  the  vector  counter 
magnetic  force  caused  by  eddy  currents.  The  latter  is  equal 
to  zero  at  the  outer  surface  of  the  wire,  and  the  net  mag- 
netic force  is  therefore  there  equal  to  the  impressed  magnetic 
force ; but  at  the  center  of  the  wire  the  counter  magnetic  force 
may  have  a value  comparable  with  the  impressed  force  and  a 
phase  nearly  in  opposition  to  it,  and  the  net  magnetic  force  may 
there  nearly  disappear.  The  value  of  the  eddy  currents  depends 
upon  the  density  of  the  inducing  magnetism,  its  frequency  of 
alternation,  and  inversely  upon  the  resistance  of  the  eddy  cur- 
rent circuits ; and  the  phase  of  the  eddy  currents  with  respect 
to  the  induced  voltage  depends  on  the  frequency  of  the  alter- 
nations of  the  magnetism  and  the  resistance  of  the  eddy  cur- 
rent circuit.  This  resistance  depends  upon  the  specific 
resistance  and  the  dimensions  of  the  conducting  body.  It 
therefore  follows  that  the  magnetic  force  of  the  eddy  currents 
may  be  relatively  large  and  its  phase  nearly  in  opposition  to 
the  impressed  magnetic  force  at  the  center  of  a thick  metal 
wire  of  low  specific  resistance,  and  the  net  magnetic  force  in 
that  case  may  be  very  small  along  the  axis  of  that  wire  al- 
though the  impressed  magnetic  force  is  large.  But  in  the  case 
of  a thin  metal  wire  of  high  specific  resistance  the  eddy  cur- 
rents must  be  more  nearly  in  quadrature  with  the  magnetic 
flux  and  small  in  volume,  and  the  net  magnetic  force  along  the 


Fig.  258.  — Eddy  Cur- 
rent in  Cross  Section 
of  Wire. 


430 


ALTERNATING  CURRENTS 


axis  of  that  wire  may  not  differ  substantially  from  that  paral- 
lel to  the  axis  near  the  surface  of  the  wire.  Since  the  resist- 
ance of  any  circular  path  of  the  eddy  currents  of  elementary 
width  and  concentric  with  the  walls  of  the  wire  is  propor- 
tional to  the  circumference  of  the  circular  path  and  therefore 
to  the  radius  of  the  path,  and  the  self-inductance  is  propor- 
tional to  the  area  of  the  circle  and  therefore  to  its  radius 
squared,  it  is  obvious  that  the  lag  angles  of  the  eddy  currents 
will  differ  along  any  radius  of  the  wire,  and  this  will  addition- 
ally cause  the  net  magnetic  force  to  differ  in  phase  from  the 
center  to  the  circumference  of  the  wire. 

In  consequence  of  these  reactions,  alternating  magnetism 
tends  to  forsake  the  central  parts  of  the  stampings  of  the  cores 
of  electrical  apparatus  unless  the  stampings  are  thin  enough  or 
of  high  enough  specific  resistance,  or  both,  to  make  the  resist- 
ance of  the  eddy  current  circuits  relatively  large.  This  results 
in  a smaller  aggregate  magnetic  flux  being  set  up  per  unit  of 
impressed  magnetic  force,  and  gives  the  appearance  of  an  in- 
crease in  the  magnetic  reluctance  of  the  core,  unless  it  is  rather 
finely  laminated. 

The  extent  of  this  effect  in  the  laminated  iron  cores  of  elec- 
trical machines  subjected  to  alternating  magnetism  was  treated 
by  Ewing  in  1891,  following  a brief  mathematical  presentation 
by  J.  J.  Thomson.* 

The  formulas  developed  by  J.  J.  Thomson,  showing  the  ex- 
tent of  magnetic  screening  in  stampings,  take  into  account  the 
effect  on  the  eddy  currents  of  the  modified  distribution  of  the 
magnetism  and  are  in  vector  form,  and  numerical  computa- 
tions are  complicated  and  laborious.  Professor  Ewing  has 
computed  a table  of  the  ratio  of  net  magnetic  force  ( 7/)  at  vari- 
ous depths  iu  a plate  to  the  magnetic  force  (770)  at  the  sur- 
face, for  iron  plates  of  various  thicknesses  when  the  frequency 
of  magnetic  alternations  is  100  cycles  per  second.  The  table  is 

* London  Electrician , Vol  28,  pp.  599  and  631.  See  also  Russell’s  Alternat- 
ing Currents , Yol.  I,  p.  359. 


HYSTERESIS  AND  EDDY  CURRENT  LOSSES 


431 


TABLE 


Thickness  of  Plate 
in  Millimeters 

:7atio  of  II  at  Middle 
to  IIq  (at  Surfaec) 

Ratio  of  ffx  to  7/0 

Ratio  of  77a  to  770 

2. 

0.120 

0.342 

0.250 

1.5 

0.245 

0.42 

0.341 

1. 

0.520 

0.629 

0.564 

0.75 

0.739 

0.82 

0.793 

0.5 

0.925 

0.940 

0.932 

0.25 

0.995 

0.995 

0.996 

here  reproduced.  The  plotted  results  for  three  thicknesses  are 
exhibited  in  Fig.  259.  In  this  figure  the  ordinates  represent 
the  ratio  of  H to  J?0,  and  the  abscissas,  measured  from  zero  either 
way,  represent  proportional  distances  from  the  center  through 
the  thickness  of  the  plate. 

It  will  be  seen  that  the 
magnetic  screening  com- 
puted by  the  formula  for  a 
plate  as  thin  as  one  half 
millimeter  (.0197  of  an 
inch)  amounts  to  approxi- 
mately 7.5  per  cent  at  the 
center,  which  is  not  of  great 
commercial  importance  in 
most  electrical  machinery. 

But  when  the  thickness  of 
the  plate  is  2 millimeters 
(.079  of  an  inch)  the  com- 
puted magnetic  screening  at 
the  center  of  the  plate  is  88 
per  cent  and  the  net  magnetic  force  at  the  center  as  com- 
puted is  only  12  per  cent  of  the  impressed  magnetic  force. 
The  figures  are  made  on  the  basis  of  iron  with  an  assumed 
uniform  magnetic  permeability  of  2000  units,  a specific  elec- 
trical resistance  of  10,000  C.  G.  S.  units  (=  10~5  ohms)  and  a 
sinusoidal  magnetizing  force  having  a frequency  of  100  periods 
per  second. 

The  symbol  R0  in  Ewing’s  table  and  charts  stands  for  the 
value  of  the  impressed  magnetic  force  at  the  surface  of  the 
plate,  Hl  stands  for  the  average  value  of  the  magnetic  force 


Fig.  259.  — Distribution  of  Net  Magnetic 
Force  across  Iron  Plates  of  Various 
Thicknesses,  when  the  Frequency  of 
Magnetic  Alternations  is  100. 


432 


ALTERNATING  CURRENTS 


over  the  cross  section  of  the  plate  at  the  instant  of  maximum 
flux,  and  H 2 stands  for  the  value  of  H which  would  produce, 
with  eddy  currents  absent,  the  maximum  total  magnetic  flux 

now  reached  in  each  period. 
Figure  260  is  a chart  taken 
from  Ewing's  article,  which 
gives  the  ratios  of  Hl  and 
to  H0  for  various  thick- 
nesses of  iron  plates. 

The  formulas  show  that 
the  effect  of  magnetic  screen- 
ing is  to  reduce  the  useful 
thickness  of  a plate  to  super- 
ficial external  layers  which 
have  a thickness  dependent 
upon 

' P_ 

Pf 


.0  0.2  0.4  0.6  0.8  1.0  1.2  1.4  1.6  1.8  2.0 

THICKNESS  OF  PLATE  IN  MILLIMETERS 

Fig.  260.  — Chart  showing  Ratios  of  Hl  and 
to  H0  in  Iron  Plates  of  Various  Thick- 
nesses. 


in  which  p is  the  specific  electrical  resistance  and  p.  the  mag- 
netic permeability  of  the  material  composing  the  plate.  The 
formulas  also  show  that,  when  the  thickness  of  the  plate  is 
large  the  eddy  current  loss  is  greatly  modified  by  the  magnetic 
screening,  and  is  proportional  to  TT02V ppf  which  is  independent 
of  the  thickness  of  the  plate.  When  the  thickness  is  small,  as 
for  iron  under  2 millimeters,  the  heating  in  each  plate  is  pro- 

portional  to  — <UzjL , where  a is  the  thickness  of  the  plate  in 

P 

millimeters.  Since  a laminated  core  is  made  up  of  many  plates 
of  thickness  a,  the  total  heating  of  a core  of  given  size  is  pro- 

portional  to  — . The  last  deduction  agrees  with  the 


deductions  reached  in  Art.  111. 

The  formulas  do  not  take  into  account  the  variation  of  per- 
meability with  the  magnetic  density  in  iron,  and  the  actual 
effect  of  the  magnetic  screening  in  cores  of  ordinary  electrical 
apparatus  is  much  less  than  the  formulas  indicate,  because  the 
permeability  goes  up  with  a reduction  of  the  magnetic  force 
when  the  magnetic  density  is  as  high  as  the  usual  densities  of 
commercial  practice,  and  the  smaller  net  magnetic  force  at  the 
center  of  a stamping  may  not  be  accompanied  by  a proportional 


HYSTERESIS  AND  EDDY  CURRENT  LOSSES 


433 


reduction  of  the  magnetism  set  up.  As  a consequence  the 
actual  effects  of  magnetic  screening  are,  doubtless,  ordinarily 
of  little  or  even  negligible  effect  in  the  cores  of  the  usual  com- 
mercial electrical  machinery,  which,  when  subjected  to  alternat- 
ing magnetism,  are  commonly  built  up  of  stampings  not  more 
than  20  mils  thick.  At  rather  low  magnetic  densities,  on  the 
other  hand,  the  permeability  decreases  when  the  magnetic  force 
is  reduced,  and  the  effect  of  magnetic  screening  may  on  this 
account  be  actually  greater  than  the  formulas  indicate. 

Figure  259  and  the  table  are  based  on  magnetic  alternations 
of  100  periods  per  second.  The  corresponding  data  for  any 
other  frequency  may  be  determined  by  the  relation  that  the 
effect  of  magnetic  screening  changes  in  approximately  direct 
proportion  to  the  first  power  of  the  frequency. 

114.  Method  for  finding  the  Form  of  Exciting  Current  required 
to  produce  a Given  Alternating  Flux  when  the  Curve  of  Magneti- 
zation is  Known.  — In  order  to  indicate  the  form  of  the  Exciting 
Current  required  to  set  up  a given  core  magnetization,  suppose 
first  an  electric  circuit  surrounding  an  iron  core  possessing  a 


Fig.  261.  — Determination  of  Exciting  Current  for  a Magnetic  Circuit  without 

Hysteresis. 

cyclic  curve  of  magnetization  like  AOB , Fig.  2(31,  which  pre- 
supposes no  appreciable  hysteresis  or  eddy  current  loss.  rlhe 
abscissas  of  this  curve  may  be  scaled  as  amperes  of  exciting 
current,  and  the  ordinates  as  magnetic  density.  Also  suppose 
that  the  form  of  the  wave  of  magnetism  in  the  core  is  to  be 
sinusoidal.  The  problem  is  to  find  the  form  of  current  wave 
that  must  flow  under  these  circumstances.  The  curve  LL  in 


434 


ALTERNATING  CURRENTS 


Fig.  261  represents  one  period  of  the  sinusoidal  wave  of  mag 
netism  in  the  core,  and  the  curve  A OB  represents  the  instan- 
taneous relations  between  the  magnetism  and  the  exciting 
current  which  sets  it  up.  The  ordinates  of  the  two  curves  are 
drawn  for  convenience  to  the  same  scale. 

To  find  the  current  required  to  set  up  the  magnetism  at  any 
instant,  such  as  corresponds  to  the  point  0 on  the  magnetism 
wave,  draw  a horizontal  line  from  0 to  I)  and  a vertical  line  from 
J)  to  I.  The  length  01  evidently  represents  the  current  in  the 
exciting  coil  which  is  required  to  set  up  the  magnetism  1)1  — CM. 
Next,  from  the  point  M,  directly  under  <7,  lay  off  a vertical 
distance  MN,  equal  to  the  current  01.  The  point  N is  one 
point  on  the  curve  of  the  exciting  current  necessary  to  set  up 
the  curve  of  magnetism  LL.  If  a large  number  of  points  on 
the  current  curve  corresponding  to  successive  points  on  the 
magnetism  wave  are  obtained  by  the  same  method,  the  result- 
ant curve  drawn  through  the  points  so  determined  is  I^I . 
The  peaked  form  of  the  exciting  current  here  indicated  is 
notable.  The  peak  is  manifestly  caused  by  the  increasing  ratio 
between  the  current  and  magnetism  as  the  magnetic  density 
becomes  greater  and  the  permeability  of  the  core  is  less.  If 
the  curve  of  magnetization  AOB  were  a straight  line,  the  ratio 
between  exciting  current  and  magnetism  would  be  constant, 
and  the  form  of  exciting  current  wave  would  then  obviously 
be  the  same  as  the  form  of  the  magnetism  wave,  the  scale  only 
being:  changed.  It  is  the  saturation  of  the  core  which  is  evi- 
deuced  by  the  crook  in  the  curve  of  magnetization  A OB  that 
causes  the  peak  in  the  wave  of  exciting  current,  and  the  degree 
of  the  peakedness  depends  on  the  degree  of  saturation  to  which 
the  core  is  forced  by  the  magnetism  set  up.  The  maximum 
exciting  current  plainly  must  come  at  the  same  instant  as  the 
maximum  magnetism,  and,  since  hysteresis  is  assumed  to  be 
negligible,  the  zero  of  current  comes  at  the  same  instant  as  zero 
magnetism. 

In  practical  cases  where  iron  forms  part  of  the  magnetic  sur- 
roundings of  the  electric  circuit,  hysteresis  is  not  negligible, 
and  the  magnetization  curve,  therefore,  takes  the  general  form 
AHBJA , Fig.  262 ; and  if  the  sinusoidal  alternating  magnetism 
wave  LL  in  the  core  is  laid  out  on  the  plot  in  the  same  manner 
as  in  Fig.  261,  the  exciting  current  1 1^  is  obtained  by  the  same 


HYSTERESIS  AND  EDDY  CURRENT  LOSSES 


435 


process  as  before ; but  the  increasing  limb  of  the  curve  of  mag- 
netization now  differs  from  the  decreasing  limb,  and  the  current 
wave  is  not  symmetrical  about  a vertical  line  drawn  through  its 
maximum  point.  A scrutiny  of  the  hysteresis  cycle  shows  that 
the  current  is  not  zero  when  the  magnetism  is  zero,  but  has  a 
magnitude  ST  = OJ  when  the  magnetism  is  passing  through  zero 
in  the  positive  direction  and  the  corresponding  value  S'  T = OH 
when  the  magnetism  is  passing  through  zero  in  the  reverse  direc- 
tion. The  current  comes  to  zero  while  the  magnetism  has  a con- 
siderable value  in  the  decreasing  quadrants  between  maximum 
and  zero  points.  For  the  hysteresis  cycle  shown  in  Fig.  262,  the 


falling  positive  current  comes  to  zero  while  the  magnetism  still 
has  the  value  ZW  = OV,  and  the  falling  negative  current  comes 
to  zero  while  the  magnetism  has  the  value  Z'W  = OV' . Since 
the  point  A on  the  hysteresis  cycle  represents  simultaneous 
maxima  of  magnetism  and  current  and  the  same  is  true  of  the 
point  B , the  maximum  values  of  the  current  occur  at  the  same 
instants  as  the  maximum  values  of  magnetism.  The  current 
wave  is  therefore  deformed  from  the  symmetry  shown  in  the  cur- 
rent curve  of  Fig.  261.  It  has  its  maximum  points  at  the  same 
instants  as  the  cyclic  curve  of  magnetism,  as  in  the  other  instance, 
but  the  zero  values  of  current  come  earlier  than  the  zero  values 
of  magnetism,  and  the  phase  of  the  exciting  current  is  advanced 
with  respect  to  the  wave  of  magnetism.  The  equivalent  angle 
of  this  advance  was  called  by  Steinmetz  the  hysteretic  angle  of 
advance.  The  fact  is,  that  the  exciting  current  now  consists  of 
two  components  : an  exciting  component  similar  to  the  current 


436 


ALTERNATING  CURRENTS 


of  Fig.  261  and.  in  phase  with  the  magnetism  ; and  an  energy- 
component  required  to  provide  the  power  absorbed  by  the  hys- 
teresis loss  in  the  core  and  the  I2B  loss  in  the  exciting  coil. 
The  latter  component  is  in  phase  with  the  impressed  voltage, 
which  is  in  leading  quadrature  with  the  magnetism  and  is  pro- 
portional to  the  rate  of  change  of  the  magnetism  at  each  instant 
in  case  the  I2H  loss  in  the  exciting  coil  is  negligible  ; but  if  the 
I2R  loss  is  not  negligible  the  phase  of  the  impressed  voltage  is 
slightly  retarded. 

In  like  manner,  the  exciting  current  required  to  set  up  sinu- 
soidal magnetism  in  a core  when  the  effect  due  to  eddy  eur- 


Fig.  263.  Exciting  Current  in  a Circuit  containing  Hysteresis  and  Eddy  Currents. 


rents  is  included  may  be  determined  by'  using  the  cyclic  curve 
of  the  iron  loss.*  The  construction  is  shown  in  Fig.  263,  where 
AFB GrA  is  a cyclic  curve  including  hysteresis  and  eddy7  currents. 
In  this  case  the  current  is  still  more  advanced.  The  component 
of  the  advanced  current  in  phase  with  the  voltage  furnishes  the 
power  which  is  absorbed  in  the  losses.  'Where  the  wave  of 
magnetism  is  sinusoidal,  as  in  Fig.  263,  the  eddy7  currents  are 
also  sinusoidal,  hence  the  component  they7  add  to  the  exciting 
current  is  sinusoidal  and  90°  in  advance  of  the  magnetism. 
This  element  of  the  exciting  current  can  be  laid  out  to  illus- 
trate its  position.  In  Fig.  263  the  dotted  curve  corres- 
ponds to  the  hysteresis  cycle  AIIBJA . and  the  curve  IfIf  is  the 
portion  of  the  exciting  current  required  to  overcome  the  mag- 
netic effect  of  the  eddy  currents.  The  ordinates  of  I1 ' I'  are 
the  algebraic  sums  of  the  ordinates  of  these  two.  It  wall  further 


* Art.  112. 


HYSTERESIS  AND  EDDY  CURRENT  LOSSES 


437 


be  shown  in  the  chapter  dealing  with  the  transformer  that  the 
eddy  currents  in  the  core  act  like  currents  in  a secondary 
winding  of  one  turn. 

115.  Relationship  between  Form  of  Magnetism  Wave  and  Form 
of  Induced  Voltage.  — In  the  foregoing  discussion,  a sinusoidal 
wave  of  magnetism  has  been  assumed.  This  may  be  readily 
produced  by  impressing  a sinusoidal  voltage  between  the  termi- 
nals of  the  exciting  coil  even  when  the  core  is  of  iron,  provided 
the  resistance  of  the  winding  of  the  coil  is  small  enough  so 
that  the  IR  drop  is  substantially  negligible.  The  impressed 
voltage,  counter  voltage  induced  by  the  magnetism  in  the  core, 
and  the  IR  drop  make  a closed  vector  triangle  ; and  if  the  IR 
drop  is  substantially  negligible,  the  counter  voltage  must  be 
substantially  equal  and  opposite  to  the  impressed  voltage.  It 
follows  from  the  relations  of  the  induced  counter  voltage  and 
the  Avave  of  magnetism  set  up  in  the  core  that  the  latter  is 
sinusoidal  when  the  former  is.  Under  the  circumstances,  the 
exciting  current  which  is  caused  by  the  sinusoidal  impressed 
voltage  to  flow  through  the  coil  inevitably  takes  on  a form  and 
volume  which  produces  sinusoidal  magnetism  of  just  sufficient 
amount  to  induce  a counter  voltage  equal  to  the  impressed 
voltage. 

Since  the  instantaneous  voltage  induced  in  a generator 
armature  or  transformer  coil  or  any  reactive  coil  Avinding  is  at 
each  instant  proportional  to  the  time  rate  of  change  of  the  mag- 
netism threaded  through  the  windings  (i.e.  is  proportional  to 

Avhere  cf>n  represents  the  magnetic  linkages  betAveen  the 
dt 

conductors  and  the  lines  of  force),  it  folloAvs  that  the  magnetic 


relations  it  may  be  seen  that  the  form  of  the  voltage  curve  may 
be  derived  from  the  curve  representing  the  change  of  magnetic 
linkages  during  a period  ; and  vice  versa , the  form  of  the  mag- 
netism Avave  may  be  derived  if  the  form  of  the  induced  voltage 
curve  is  knoAvn.  For  instance,  if 


cj),,  = A1  sin  (a  + /3j)  + A2  sin  (2  a + /32)  + A3  sin  (3  « + /S3) 

+ •••  An  sin  (ri«  + /3„), 


linkages  at  each  instant  are  proportional 


From  these 


438 


ALTERNATING  CURRENTS 


then,  sines  da  = 2 nfdt , 

e = — cos  («  + /Sj)  + 2 A2  cos  (2  a + yS2) 

+ 3 A3  cos  (3  a + /S3)  + b nA.„  cos  (/?«  + /?„)]. 

2 7r/* 

In  other  words  the  ratio  — r ~ is  introduced  into  the  ordinates 

108 

of  the  curve  of  voltage,  all  the  harmonics  are  moved  one 
quarter  period,  and  the  curve  of  voltage  is  therefore  in  quadra- 
ture with  the  curve  of  the  magnetic  linkages,  and  the  ordinates 
of  each  harmonic  in  the  voltage  curve  are  affected  by  a coeffi- 
cient equal  to  the  ratio  of  the  frequency  of  the  particular  har- 
monic to  the  fundamental  frequency.  Every  harmonic  in  the 
wave  of  magnetic  linkages  enters  into  the  wave  of  induced  volt- 
age, and  the  ordinates  of  the  harmonics  are  exaggerated  in  the 
voltage  wave  in  comparison  with  the  ordinates  of  the  harmonics 
of  the  wave  of  magnetism,  in  an  order  proportional  to  the  ratio 
of  their  several  frequencies  compared  with  the  fundamental  fre- 
quency. It  is  plain  from  this  that  induced  voltage  of  sinu- 
soidal form  cannot  be  derived  from  magnetic  linkages  varying 
according  to  any  other  function.  Any  deviation  of  the  mag- 
netic wave  from  the  sine  form  produces  a greater  deviation 
in  the  induced  voltage. 

In  the  case  of  a transformer  or  similar  coil  under  conditions 
of  negligible  magnetic  leakage,  the  number  of  magnetic  linkages 
at  each  instant  is  equal  to  the  number  of  turns  in  the  coil  multi- 
plied by  the  number  of  lines  of  force  in  the  aggregate  magnetic 
flux  at  that  instant ; but,  in  the  case  of  alternator  armatures 
or  other  windings  which  are  distributed  or  are  affected  In- 
appreciable magnetic  leakage,  the  number  of  magnetic  linkages 
at  any  instant  is  not  equal  to  the  product  of  turns  in  the  wind- 
ing by  instantaneous  flux,  but  the  instantaneous  summations  of 
flux  embraced  by  the  individual  turns  must  be  used  as  the 
ordinates  of  the  magnetism  wave  when  plotting  that  curve. 

116.  Alternating  Flux  caused  by  a Current  of  Predetermined 
Form.  — The  wave  of  current  in  the  main  wire  of  a circuit  is 
the  geometrical  resultant  of  the  current  in  all  the  branches  of 
the  circuit.  An  inductive  coil  placed  in  series  with  the  line 
will  have  a current  flow  through  it  which  is  dependent  for  its 
form  upon  the  characteristics  of  all  parts  of  the  circuit.  Sup- 


HYSTERESIS  AND  EDDY  CURRENT  LOSSES 


439 


pose  for  convenience  that  the  voltage  impressed  between  the 
terminals  of  a circuit  which  contains  an  impedance  coil,  among 
other  things,  is  of  such  a form  that  a sinusoidal  current  is 
caused  to  flow  through  the  circuit.  This  fixes  the  form  of  the 
current  in  the  winding  of  the  impedance  coil.  The  form  of 
the  magnetism  wave  set  up  in  the  core  of  the  impedance  coil  bj 
the  current  in  its  winding  may  be  obtained  by  reversing  the 
procedure  set  forth  in  Art.  114,  if  eddy  currents  are  negligible. 
The  construction  is  shown  in  Fig.  264  for  a core  containing 
hysteresis  but  neglecting  eddy  currents.  Here  curve  II  repre- 
sents a sinusoidal  current  passing  through  the  impedance  coil, 


Y 


Fig.  264. — Wave  of  Magnetic  Flux  caused  by  Sinusoidal  Current. 


and  AHBJA  is  the  cyclic  hysteresis  curve.  Taking  any 
instantaneous  value  of  the  current,  such  as  MN,  laying  it  off  on 
the  axis  of  abscissas  from  0,  as  01,  the  ordinate  of  the  hysteresis 
cycle  erected  from  I to  the  hysteresis  cycle  at  D shows  the 
magnetic  density  corresponding  to  current  MN.  The  point  C 
at  the  intersection  of  the  horizontal  line  drawn  from  D with 
the  line  MN  extended,  is  one  point  on  the  wave  of  magnetism. 
The  curve  representing  the  wave  of  magnetism  may  be  deter- 
mined for  an  entire  cycle  by  continuing  this  process.  The 
curve  LL  in  Fig.  264  shows  the  wave  of  magnetism  set  up  by 
the  current  II  in  the  core  with  cjmlic  curve  of  magnetization 
AHBJA,  assuming  the  latter  to  be  plotted  in  magnetic  densities 
for  ordinates  and  amperes  flowing  in  the  exciting  coil  for 
abscissas. 

The  induced  counter  voltage  in  the  coil  is  a very  irregular 
curve  under  the  circumstances  here  considered.  It  is  propor- 


440 


ALTERNATING  CURRENTS 


tional  at  each  instant  to  the  time  rate  of  change  of  the  magnetism. 
Since  the  maximum  of  magnetism  comes  at  the  same  point  as  the 
maximum  of  exciting  current,  the  induced  counter  voltage 
passes  through  zero  at  the  instant  the  exciting  current  passes 
through  its  maximum.  It  will  be  observed  that  the  eddy  cur- 
rents in  the  core  depend  upon  the  induced  voltage  in  circuits 
surrounding  the  magnetism,  and  the  eddy  current  cycle  cannot 
be  assumed  to  be  the  same  as  one  caused  by  sinusoidal  eddy 
currents. 

The  considerations  of  this  article  are  especially  applicable  to 
the  study  of  current  transformers  used  with  amperemeters  and 
wattmeters. 

117.  Effect  of  Iron  Losses  on  the  Apparent  Resistance  and  Re- 
actance of  a Circuit.  — The  foregoing  discussion  of  iron  losses 
leads  to  a generalization  of  great  importance. 

The  true  electrical  Resistance  of  a conductor  is  a quantity 
which  depends  solely  on  the  dimensions,  temperature,  and  char- 
acter of  the  conductor.  It  is  the  electrical  resistance  which  is 
measured  by  means  of  a Wheatstone  bridge,  using  continuous 
currents  in  the  process.  It  is  the  same  whether  an  alternating 
current  or  a direct  current  flows  through  the  circuit,  though  it 
may  be  masked  by  various  effects  of  alternating  currents. 

In  a perfectly  insulated  circuit  carrying  continuous  currents, 
the  causes  of  losses  of  energy  lie  in  the  passage  of  the  currents 
through  the  resistances  of  the  conductors  composing  the  circuit 
and  the  currents  through  devices  which  absorb  energy  by 
presenting  a counter  voltage.  In  the  case  of  alternating  cur- 
rent flow,  additional  sources  of  loss  of  energy  become  evident. 
Energy  may  be  absorbed  by  the  effects  of  hysteresis  and  eddy 
currents.  The  hysteresis  may  be  either  magnetic  hysteresis  or 
dielectric  hysteresis.  An  effect  also  arises  from  the  self-induc- 
tion of  the  conductor,  which  causes  a concentration  of  the 
current  near  the  surface  of  the  conductor,  which  is  treated  in 
a later  chapter  under  the  name  of  skin  effect. 

Leakage  effects  due  to  imperfect  insulation,  and  the  effects  of 
convection  accompanying  brush  discharges  and  the  development 
of  a luminous  corona  between  conductors  of  the  circuit  at  very 
high  potentials,  occur  in  either  continuous  current  or  alternating 
current  circuits.  These  effects  also  result  in  losses  of  energy. 

The  angle  of  lag  in  an  alternating  current  circuit  is  obtained 


HYSTERESIS  AND  EDDY  CURRENT  LOSSES 


441 


from  the  expression  9 = cos-1 


where  P is  the  power  meas- 


ured by  wattmeter  and  El  is  the  volt-amperes  obtained  by 
multiplying  current  by  voltage.  This  angle  6 is  the  angle  sub- 
tended between  Z and  R in  the  right-angled  triangle  defining 
the  relations  of  Z,  R , and  X,  namely : 


operator  Z = R - f jX,  and 
tensor  Z—  V R2  + X 2. 


From  this  triangle  also  follows  the  right-angled  triangle 
IZ  = X — IR  + jlX  ==  Er  + jEx. 


In  this  instance,  the  angle  subtended  between  E and  IR  is  the 
angle  9.  The  value  of  X in  these  triangles  is  determined  by 
the  frequency  of  the  current  and  the  characteristics  of  the  cir- 
cuit which  fix  its  inductance  and  capacity,  and  its  value  is  not 
affected  by  the  power  transferred  through  the  circuit.  The 
question  at  issue  is:  Does  the  value  of  R (in  ohms)  of  the 
horizontal  sides  of  these  triangles  correspond  with  the  circuit 
resistance  as  measured  by  a Wheatstone  bridge?  A consid- 
eration of  the  relations  of  the  sides  of  the  triangles  results  in  a 


negative  answer.  Since  the  horizontal  side  of  the  impedance 

p 

triangle  must  be  equal  to  Z cos  6 = Z — — and  E = Zf  it  follows 

p m pp 

that  Z cos  6 = — , which  will  always  differ  from  R = ^ if  any 


energy  is  expended  in  the  circuit  besides  the  I2R  loss.  This 
points  the  way  to  the  generalization  of  the  term  resistance  in 
which  it  is  referred  to  as  either  positive  or  negative  in  connec- 
tion with  the  semicircle  diagrams  of  Art.  70. 

Since  power  may  be  either  positive  or  negative,  that  is,  it 
may  either  be  absorbed  by  and  expended  in  a circuit  or  be  gen- 

P 

erated  in  and  delivered  by  the  circuit,  it  is  obvious  that  -^like- 
wise may  be  positive  or  negative,  since  its  sign  depends  on  P. 
In  general,  then.  Equivalent  resistance  of  a circuit  may  be  de- 
fined as  the  ratio  of  power  expended  in  or  by  a circuit  to  the 
square  of  the  current  flow.  When  an  alternating  current  flows 
through  a circuit,  the  equivalent  resistance  ordinarily  differs 


442 


ALTERNATING  CURRENTS 


from  the  electrical  resistance  of  the  circuit  as  measured  by  a 

P — I2R 

Wheatstone  bridge  by  an  amount  equal  to — — where  Pis 


the  wattmeter  reading  and  R is  the  electrical  resistance  of  the 
circuit. 

When  the  word  resistance  is  used  in  this  text  in  relation  to 
alternating  current  circuits,  it  may  be  taken  as  meaning  the 
equivalent  resistance  unless  it  is  qualified  by  the  word  true  or 
the  word  electrical.  The  equivalent  resistance  is  the  horizontal 
component  of  the  impedance  triangle. 

In  any  circuit  carrying  a steady  current,  the  equivalent  resist- 
ance and  the  true  resistance  are  always  equal  unless  the  effects 
of  some  voltage-generating  device  are  included  within  the  circuit. 

As  a comparison  of  equivalent  resistance  with  the  true  resist- 
ance, their  values  for  the  primary  coil  of  a transformer  may  be 
taken  when  the  secondary  circuit  of  the  transformer  is  open 
and  therefore  does  not  absorb  any  power.  In  the  case  of  a cer- 
tain transformer,  the  voltage  was  1100  volts,  the  exciting  cur- 
rent .2  amperes,  and  the  wattmeter  reading  which  gives  the 
power  expended  in  hysteresis  and  eddy  currents  in  the  core  and 
I2R  in  the  winding  150  watts.  From  these  figures,  the  equiva- 
lent resistance  of  the  winding  is  shown  to  be  3750  ohms.  The 
impedance  is  5500  ohms.  The  electrical  resistance  of  the  con- 
ductor composing  the  winding  was  only  one  and  a half  ohms. 
In  another  instance  of  similar  kind  the  voltage  was  1100,  the 
exciting  current  .043  amperes,  and  the  wattmeter  reading  37 
watts.  This  gives  an  equivalent  resistance  of  20,000  ohms  and 
an  impedance  of  25,600  ohms.  The  electrical  resistance  was  7 
ohms.  In  another  instance  the  voltage  was  60  volts,  the  cur- 
rent .86  amperes,  and  the  power  43.3  watts.  This  gives  equiv- 
alent resistance  of  58  ohms  and  impedance  of  70  ohms.  The 
electrical  resistance  of  the  conductor  was  .08  ohms. 

It  is  obvious  that  the  power  consumption  in  the  iron  cores  of 
these  transformers  enlarges  the  power  factor  by  advancing  the 
current  to  a phase  nearer  the  voltage.  This  is  partially  caused 
by  hysteresis  and  partially  by  eddy  currents.  The  part,  as  stated 
before,  caused  by  hysteresis  was  called  the  “ hysteretic  angle  of 
advance”  by  Professor  Steinmetz.  This  is  an  angle  of  advance 
or  negative  angle  of  lag,  and  is  not  the  whole  of  the  effect  pro- 
duced by  the  iron  core,  though  it  is  often  the  greater  part  of  it. 


CHAPTER  X 


MUTUAL  INDUCTION,  TRANSFORMERS 

118.  Mutual  Induction  and  Elementary  Transformer  Diagrams. 

— The  remarkable  development  in  the  use  of  alternating  cur- 
rents for  transmitting  and  distributing  electric  power  is 
mainly  due  to  the  facility  and  economy  with  which  they  may 
be  transformed  from  one  voltage  to  another.  The  induction 
coils  that  are  used  for  this  purpose  are  called  transformers,  as 
already  explained,  and  their  action  is  due  to  the  inductive 
effect  which  a varying  current  in  one  circuit  exerts  upon  an 
adjacent  circuit  as  well  as  upon  the  circuit  itself.*  Since  this 
effect  is  a mutually  interacting  one,  it  is  called  Mutual  induction. 
Evidently,  if  the  magnetism  due  to  a flow  of  current  in  one 
coil,  which  may  be  called  the  primary  coil,  see  Fig.  46,  passes 
through  the  turns  of  another  coil,  which  may  be  called  the  sec- 
ondary coil,  a variation  of  the  current  in  the  primary  coil  sets 
up  a voltage  in  the  secondary  coil.  This  voltage,  when  a sinu- 
soidal alternating  current  passes  through  the  first  coil,  is  90° 
behind  the  phase  of  the  current  in  the  primary  coil,  when  core 
losses  in  the  magnetic  circuit  are  negligible.  This  is  also  true 
of  self-inductive  voltage,  and  for  the  same  reasons. f In  case 
the  external  electric  circuit  of  the  secondary  coil  is  open  and  the 
magnetism  in  the  magnetic  circuit  is  sinusoidal  and  all  passes 
through  both  coils,  a phase  diagram  similar  to  Fig.  114  may  he 
constructed  to  represent  the  phase  relations  of  the  primary  cur- 
rent, magnetic  flux,  impressed  voltage,  and  primary  and  second- 
ary induced  voltages.  Such  a diagram  is  exhibited  in  Fig.  265, 
in  which  OF  represents  the  impressed  voltage  Fx ; 01  the  current 
in  the  primary  coil  Ix ; OF , the  self-induced  voltage  in  the  pri- 
mary coil  Flm',  and  OK  the  induced  voltage  of  the  secondary 
coil  Ev  In  this  case  the  ratio  of  OF  to  OK  is  evidently  equal 
to  the  ratio  of  the  numbers  of  turns  in  the  two  coils,  since  it  is 

t Art.  37, 


* Art.  23. 


442 


444 


ALTERNATING  CURRENTS 


assumed  that  all  of  the  magnetic  flux  links  both  coils,  and 
there  is  no  secondary  resistance  drop,  since  no  current  flows. 
OC  represents  the  IR  drop  in  the  primary  coil,  and  the  primary 
ampere-turns  n1Il  are  proportional  to  01.  The  magnetic  flux 
is  assumed  to  be  in  phase  with  the  current.  Had  iron  losses 

not  been  neglected  the  angle  6X  would 
have  been  less,  as  the  active  com- 
ponent of  01  would  then  have  been 
required  to  furnish  these  as  well  as 
the  I12R1  losses.  Also  the  angle  be- 
tween 01  and  OF  would  have  ex- 
ceeded 90°  by  the  hysteresis  angle 
of  lead,  but  the  magnetism  itself 
would  still  remain  in  quadrature  re- 
lation with  E'lm.  In  this  and  the 
discussion  immediately  following  the 
subscript  m is  used  to  denote  in- 
duced voltages  caused  by  variable 
magnetic  fluxes  which  pass  through 
both  primary  and  secondary  coils. 
When  the  letter  E is  used  to  repre- 
sent primary  voltages  it  is  given  a 
prime  when  representing  induced 
voltages,  and  is  without  the  prime 
when  representing  drops  of  impressed 
voltages. 

Thus  far  the  effect  of  mutual  in- 
duction is  similar  to  that  of  self-in- 
duction. But  when  a current  flows 
in  the  secondary  coil  under  the  in- 
fluence of  the  mutually  induced  vol- 
tage, another  effect  is  produced, 
which  is  not  so  closely  related  to 
the  conditions  in  a self-inductive 
circuit.  The  current  in  the  second- 


Fig. 265.  — Phase  Diagram  of 
Voltage  and  Current  in  a 
Transformer  having  Negli- 
gible Iron  Losses  with  the 
Secondary  Circuit  Open. 


ary  coil  tends  to  demagnetize  the  magnetic  circuit  apper- 
taining jointly  to  the  two  coils.  But  the  induced  voltage  in 
the  primary  coil  must  at  each  instant  be  equal  to  the  alge- 
braic difference  between  the  values  of  the  active  and  impressed 
voltages  at  that  instant ; that  is,  the  instantaneous  impressed 


MUTUAL  INDUCTION,  TRANSFORMERS 


445 


voltage  must  be  equal  to  the  algebraic  sum  of  the  corre- 
sponding instantaneous  active  and  reactive  voltages  (the 
components  of  impressed  voltage  drop  in  primary  resistance 
and  reactance),  the  latter  being  equal  to  the  self-induced 
voltage  in  scalar  value  but  reversed  in  direction ; or  ex 
— elm  + e1R^  where  elm  is  the  primary  reactive  voltage  and  e1R^ 
is  the  active  voltage  which  drives  the  primary  current  through 
the  primary  resistance.  If  the  waves  of  voltage  are  sinusoidal, 
we  may  write  Elm  = V Ex2  - E1R2  or  Ex  = VE1R2  + Elm2,  in 
which  expressions  Ex  is  the  effective  value  of  impressed  vol- 
tage, E1R  of  active  voltage,  and  Elm  of  reactive  voltage.  This 
is  a fixed  relation  determined  by  the  law  that  the  algebraic 
sum  of  instantaneous  voltages  taken  around  a circuit  must  re- 
duce to  zero,  and  is  therefore  independent  of  the  amount  of 
current  flowing  in  either  the  primary  or  secondary  circuit. 
Then  if  the  characteristics  of  the  secondary  circuit  (load  on 
the  secondary)  are  changed  so  that  more  secondary  current 
flows  with  a resultant  decrease  of  induced  voltage  below  the 
value  shown  by  the  relation  elm  = ex  —e1Ri,  an  increased  primary 
current  flows  in  order  to  maintain  the  relation ; and  if  the 
characteristics  of  the  secondary  circuit  are  changed  so  as  to 
decrease  the  secondary  current  with  a resultant  increase  of  the 
induced  voltage  as  indicated  by  the  equation  elm  = ex  — eXRl,  the 
primary  current  falls,  so  as  to  reestablish  the  relation.  In  other 
words,  if  the  magnetic  flux  which  induces  EXm  is  disturbed, 
as  by  introducing  a magnetizing  effect  extraneous  to  the  pri- 
mary coil,  such  as  is  caused  by  current  flowing  in  the  second- 
ary coil,  the  primary  current  will  again  adjust  itself  so  as  to 
maintain  the  inducing  flux  at  such  value  as  still  to  produce 
the  relation  eXm  = ex  — e1R  . Consequently,  the  magneto-motive 
force  of  the  primary  and  secondary  currents  of  a transformer 
must  so  combine  at  each  instant  that  the  resultant  magneto- 
motive force  will  create  a wave  of  magnetic  flux  in  the  core, 
the  rate  of  change  of  whose  linkages  with  the  turns  of  the 
primary  coil  is  equal  to  108  times  the  primary  self-induced  vol- 
tage, which  voltage  must  in  turn  be  at  each  instant  equal  to 
the  difference  between  the  impressed  and  active  voltages  at 
that  instant,  witli  algebraic  sign  reversed.  In  the  special  case 
where  the  voltages  are  sinusoidal,  the  two  magneto-motive 
forces  must  combine  in  such  a way  as  to  create  a sinusoidal 


446 


ALTERNATING  CURRENTS 


wave  of  flux  which  is  advanced  90°  ahead  of  the  wave  of  self- 
induced  voltage  and  is  of  a magnitude  that  produces  a vector 
of  self-induced  voltage  equal  to  — — E1R^)  = E1Ri  — Er 

The  relation  of  impressed,  active,  and  reactive  voltage  then  is 
Ex  — ElRi  — E]jn  — 0.  The  vector  OElm ! of  Fig.  265  is  equal 
to  - Elm. 

This  means  that  when  the  secondary  circuit  is  closed  and  a 
current  flows  under  the  impulsion  of  the  secondary  induced 
voltage,  the  primary  current  must  take  such  magnitude  and 
phase  position  as  will  in  conjunction  with  the  secondary  am- 
pere-turns produce  the  requisite  magneto-motive  force  to  act 
in  the  joint  magnetic  circuit.  This  further  means  that  the  pri- 
mary ampere-turns  must,  at  each  instant,  neutralize  the  effect  of 
the  secondary  ampere-turns  in  the  joint  magnetic  circuit,  and 
must  in  addition  supply  the  ampere-turns  which  maintain  the 
inducing  flux,  i.e.  nli1  — n2i2-\-  , where  the  terms  in  the 

equation  represent  respectively  corresponding  instantaneous 
values  of  the  total  primary  ampere-turns,  the  ampere-turns 
of  the  secondary  circuit,  and  the  resultant  ampere-turns  of  the 
primary  and  secondary  coils.  It  follows  from  this  that  il  = 
1 n 

- i2  + i/,  in  which  s = -1,  nx  and  n2  being  respectively  the  num- 
8 »2 

bers  of  turns  of  conductors  comprised  in  the  primary  and  sec- 
ondary coils.  Since  flux  equals  magneto-motive  force  divided 
by  reluctance,  and  the  reluctance  of  an  iron  core  varies  with 
magnetic  density,  the  exciting  current  ij  varies  quite  irregu- 
larly in  a transformer  with  an  iron  core,  even  though  the  im- 
pressed voltage  and  magnetic  flux  vary  sinusoidally.  The 
total  primary  current  which  flows,  when  the  secondary  circuit 
is  open,  multiplied  by  the  primary  coil  turns  may  be  called  the 
Exciting  ampere  turns. 

If  the  magneto-motive  force  of  either  or  both  coils  sets  up 
flux  which  does  not  pass  through  the  other  coil,  the  effect  is 
similar  to  the  introduction  of  external  self-inductance  into  the 
circuits  of  the  respective  coil  or  coils  in  addition  to  the  mutual 
inductance,  as  such  fluxes  surround  only  the  currents  producing 
them.  Such  flux  is  usually  termed  Magnetic  leakage.  In  cases 
where  there  is  no  magnetic  leakage  or  other  variable  flux  link- 
ing only  the  secondary  coil  or  its  external  circuit,  the  phase  of 
the  secondary  current  is  coincident  with  the  phase  of  the  induced 


MUTUAL  INDUCTION,  TRANSFORMERS 


447 


voltage  in  the  secondary  coil.  Then  if  the  primary  current, 
the  exciting  element  of  the  primary  current,  and  the  secondary 
current,  are  assumed  to  he  sinusoidal,  and  there  are  no  core 
losses  in  the  magnetic  circuit,  as  may  be  the  case  if  the  mag- 
netic circuit  includes  no  iron,  the  relation  becomes  Mx  = 
Vff/j2  4-  Tf22,  where  the  letters  Mv  and  A f2  represent  the 
scalar  values  of  magneto-motive  forces  set  up  in  the  joint  mag- 
netic circuit  by  the  currents.  (In  this  early  discussion  of  the 
transformer  it  is  assumed  that  effects  of  electrostatic  capacity 
are  absent.)  The  vector  relation  is  then  = Mx  — pro- 
vided no  magneto-motive  forces  act  in  the  magnetic  circuit 
except  those  set  up  by  the  currents  mentioned. 

A simple  diagram,  using  equivalent  sinusoids  for  currents  and 
voltages,  such  as  is  shown  in  Fig.  266,  develops  these  relations 
graphically.  Vectors  Ex(  = OE)  and  OA)  represent  the 

impressed  voltage  and  exciting  ampere-turns  (the  latter  exag- 
gerated in  length  for  the  sake  of  clearness)  of  a transformer 
having  negligible  leakage  and  a non-reactive  secondary  circuit. 
The  vector  nx  1^  and  the  vector  OB  representing  total  primary 
ampere-turns  may  be  vectors  scaled  to  also  represent  either 
primary  currents  measured  in  amperes  or  magneto-motive  forces 
measured  in  gilberts,  while  the  secondary  vector  of  ampere-turns 
(9(7,  by  using  the  same  scales,  equals  the  scalar  value  and  posi- 
tion of  secondary  current  divided  by  the  ratio  of  transformation, 
s,  or  the  gilberts  of  secondary  magneto-motive  force.  The 
vector  nxIJ(  = Off)  represents  the  equivalent  magnetizing  ele- 
ment of  the  exciting  magneto-motive  force  and  lags  behind  the 
latter  by  an  angle  equal  to  the  angle  of  advance  caused  by  the 
iron  losses  in  the  core.  It  is  90°  in  the  lead  of,  and  determines 
the  direction  of,  the  primary  and  secondary  induced  voltages, 


tors  is  laid  out  the  secondary  vector  of  magneto-motive  force 
n2I2(  = (9(7),  the  secondary  cii’cuit  here  being  assumed  non- 
reactive. The  vectors  n2I2  and  nxI^  must  then,  to  conform  to 
the  results  of  the  discussion  above,  form  a closed  vector  triangle 
with  the  total  magneto-motive  force  of  the  primary  current 
ttj/p  or  nxlx  and  n2I2  must  combine  to  furnish  the  resultant 
magneto-motive  force  acting  in  the  joint  magnetic  circuit. 
On  the  line  OB , nxIv  a distance  (9(7  is  laid  off,  which  represents 


On  the  line  of  the  latter  vec- 


448 


ALTERNATING  CURRENTS 


Y 


Fig.  266.  — Phase  Diagram  of  Voltages 
and  Currents  in  a Transformer  when  a 
Current  flows  in  the  Secondary  Circuit. 
The  various  vector  quantities  used  in  the 
construction  are  represented  by  lines  as 
follows:  Impressed  voltage  E1  by  OE; 
primary  self-induced  and  secondary 
mutually  induced  voltages  E\m  and 
sE'om  by  OF;  voltage  drop  in  resist- 
ance of  primary  coil  E1R  by  OG;  excit- 
ing ampere-turns  n i/M  and  current  I ^ by 
0 A ; secondary  ampere-turns  ?ioX>  and 

currenti/j by  OC  ; quadrature  ampere- 

a 


that  portion  of  the  voltage  E^ 
which,  multiplied  by  the  cur- 
rent Iv  furnishes  the  necessary 
power  to  drive  the  primary 
current  through  the  primary 
resistance,  or  E1R  . The  vol- 
tage E1  minus  this  voltage  Elf^ 
must  be  numerically  equal  but 
opposite  to  the  primary  self- 
induced  voltage  E1'(=  — OF) 
and  also  numerically  equal 
but  opposite  to  the  mutually 
induced  secondary  voltage 

E2m  = — (^OF^j  multiplied 

the  ratio  of  transformation,  s. 

The  triangle  ODE  of  Fig. 
266  corresponds  to  the  tri- 
angle OCE  in  Fig.  265  and 
OCA  in  Fig.  114,  i.e.  OD, 
OC , and  OC  are  the  respec- 
tive components  of  impressed 
voltage  projected  on  the  cur- 
rent, and  DE,  CE , and  CA 
are  the  respective  components 
in  quadrature  to  the  current ; 
but  in  the  case  of  Fig.  266, 
unlike  the  other  two,  onl}T 
part,  E1Ri , of  the  voltage  is 
lost  by  reason  of  power  used 
in  the  primary  circuit.  The 
remaining  power  represented 
by  the  vector  product  of  El  II 
is  transferred  to  the  magnetic 
circuit,  and  thence,  deducting 
core  losses,  to  the  secondary 
electric  circuit.  Therefore,  it 
is  evident  that  the  effect  of 


turns  nil/  by  OH;  primary  current 
and  ampere-turns  nj/j  by  OH. 


the  current  in  the  secondary 
circuit  has  been  to  decrease 


MUTUAL  INDUCTION,  TRANSFORMERS 


449 


E 

— 1 of  the  primary  circuit, 

h 

X' 

and  the  angle  of  lag,  6X  = tan-1  ■ , where  Xx  and  Rx  are  the 

-“'l 

apparent  primary  reactance  and  resistance.  Starting  with 
nxIx  = nxI^  as  shown  in  Fig.  265,  for  a transformer  with  open 
secondai’y  circuit,  then  closing  the  secondary  circuit  and  gradu- 
ally reducing  its  resistance  until  full  load  is  reached,  will  cause 
the  vectors  and  nxIv  in  a commercial,  constant  voltage 
transformer,  quickly  to  become  so  extended  in  length  that 
the  parallelograms  of  magneto-motive  forces  and  voltages 
will  have  flattened  out  so  much  that,  for  most  practical  pur- 
poses, the  primary  current  and  secondary  current  may  be 
considered  to  differ  substantially  180°  in  phase.  Thus,  in  a 
certain  transformer  of  20  kw.  capacity,  the  exciting  current  is 
about  one  half  of  an  ampere,  while  the  full  load  primary 
current  is  ten  amperes,  or  Ix  is  twenty  times  larger  than  1 ^ a 
difference  which  is  still  more  marked  in  transformers  of  larger 
sizes.  Therefore,  in  a transformer  without  appreciable  mag- 
netic leakage,  and  fully  loaded  with  a non-reactive  load,  the 
apparent  self-inductance  almost  disappears,  and  the  primary 
and  secondary  act  like  a single  circuit  made  up  almost  entirely 
of  non-reactive  resistance.  The  apparent  primary  impedance 
reduces  to  nearly  the  sum  of  the  resistances  of  the  primary  coil 
and  equivalent  resistance  of  the  secondary  coil  and  the  sec- 
ondary external  load  circuit.  There  is  very  little  variation  in 
either  the  phase  position  or  scalar  value  of  1^  with  change  of 
load,  as  will  be  seen  later.* 

The  various  conditions  and  problems  that  arise  in  the  opera- 
tion of  transformers  are  entered  into  extensively  in  later  articles, 
hut  before  taking  up  the  special  cases  of  sinusoidal  or  other 
periodic  impressed  alternating  voltages,  we  will  consider  the 
more  general  case  of  the  characteristics  of  mutually  reactive 
coils  as  they  are  affected  by  any  change  of  currents  or  voltages 
whatever. 

119.  Mutual  Inductance. — Consider  two  adjacent  coils  sur- 
rounded by  air  and  in  which  currents  are  flowing.  Then  at 
any  instant  the  total  number  of  linkages  of  lines  of  force  with 
the  turns  of  the  conductors  composing  either  one  of  the  coils,  is 


both  the  apparent  impedance  Z x 


2g 


* Art.  128. 


450 


ALTERNATING  CURRENTS 


the  number  of  linkages  due  to  the  current  in  that  coil  alge- 
braically added  to  the  number  of  linkages  due  to  the  other  coil 
which  are  embraced  by  the  first  coil.  If  the  current  is  changed 
in  either  of  the  coils,  an  instantaneous  voltage  equal  to  ej 

_ _ is  induced  in  the  coil  under  consideration,  which 

108  dt 


may  be  called  the  primary  coil,  where  nx  is  the  number  of  turns 
and  (p1  the  instantaneous  average  magnetic  flux  linked  per 
turn  of  the  coil.  The  average  number  of  magnetic  linkages 
per  turn  with  the  turns  of  the  coil  due  to  its  own  current  is 

-^ij^  in  which  Lx  and  I are  respectively  the  self-inductance 

ni 

and  the  current  in  the  coil.* 

The  number  of  lines  of  force  due  to  the  current  in  the  first 
coil  which  pass  through  the  second  coil  evidently  depends  upon 
the  relative  positions  of  the  coils,  but  it  cannot  be  greater  than 
the  total  number  of  lines  set  up  by  the  current  in  the  first  coil. 
For  any  two  fixed  coils  in  a medium  of  constant  permeability, 
this  number  of  lines  is  proportional  to  the  current  flowing  in  the 
primary  coil.  The  voltage  developed  in  the  second  coil,  due 

to  a change  in  the  current  flowing  in  the  first,  is  eJ  = — 

° & 2 108cft 


where  «2  and  <£2  are  the  turns  in  the  second  coil  and  the  instan- 
taneous average  magnetic  flux  linked  therewith  per  turn  of  that 

coil.  The  equation  e-J  = — may  pe  written  eJ  = — 

^ 1 108  dt  J 1 dt 


when  the  permeability  is  constant.*  The  equation  e2' = 

— may  be  similarly  written  eJ  = — — — 1 where  108J/ 

!U8(ft  J J 2 dt 


is  the  number  of  linkages  with  the  turns  of  the  second  coil  of 
lines  of  force  which  are  due  to  the  first  coil  when  one  ampere 
is  flowing  in  the  first  coil,  supposing  the  coils  to  be  in  air  or 
other  medium  of  unchanging  permeability.  The  expression 
represented  by  M is  called,  by  analogy,  the  Mutual-inductance 
or  the  Coefficient  of  mutual  induction  of  the  coils.  If  k is  a 
coefficient  numerically  equal  to  the  ratio  of  the  reluctance  of 
the  path  of  the  lines  of  force  which  interlink  the  two  coils  to 
the  aggregate  reluctance  of  the  path  of  all  the  lines  of  force  set 
up  by  the  primary  coil,  then  the  value  of  cf)2k  is  equal  to  <f>v 

* Art.  41. 


MUTUAL  INDUCTION,  TRANSFORMERS 


451 


If  the  coils  are  long  solenoids  in  air,  with  current  only  in  the 
first  coil,  the  flux  in  the  first  coil  is  (f>l  = where  A is 


the  cross  section  and  l the  length  of  the  coil ; and  if  the  sole- 
noids are  wound  one  over  the  other  so  that  their  dimensions 
are  practically  equal,  the  value  of  k is  unity,  because  02  evi- 
dently becomes  equal  to  <f>v  and  M becomes  equal  to  . 
If  the  same  current  is  now  switched  into  the  second  coil,  we 

have  <A  = ^ rTn^^  aiKl  M'  ==  But  in  this  special  case 

, 10 1 108/  1 , 

= and  hence  M=M '.  Also  Lx  = ^ and  = Ml  . 
<£2  n2  1 108/  2 108/ 

Consequently  = M 2,  or  M — V LXLV 

If  k is  not  equal  to  unity,  it  is  manifest  that  some  of  the 
magnetic  flux  set  up  by  the  ampere-turns  of  each  coil  do  not 
link  with  the  turns  of  the  other  coil.  Such  lines  of  force  are 
called  Magnetic  leakage  or  Leakage  flux. 

If  the  foregoing  long  solenoids  are  now  separated  by  draw- 
ing one  out  of  the  other,  the  value  of  M continually  decreases 
as  the  separation  continues,  because  the  paths  of  the  magnetic 
flux  set  up  by  the  two  coils  become  differentiated.  The  self- 
inductances of  the  coils  remain  constant,  so  that  as  the  coils 
separate,  the  mutual-inductance  becomes  less  than  V LxLr  As 
the  separation  of  the  coils  becomes  greater,  the  value  of  M de- 
creases towards  a minimum  of  zero.  It  reaches  the  latter  value 
when  the  coils  are  at  an  infinitely  great  distance  apart  or  the 
axis  of  one  is  placed  symmetrically  but  at  right  angles  with 
reference  to  the  axis  of  the  other,  so  that  the  summation  of  the 
linkages  of  the  magnetic  flux  from  each  coil  through  the  turns 
of  the  other  reduces  to  zero. 

The  maximum  possible  value  of  the  mutual-inductance  of  the 
two  coils  is  therefore  a mean  proportional  between  the  values 
of  their  self-inductances,  and  the  minimum  value  is  zero.  The 
maximum  value  can  only  be  attained  when  all  the  lines  of  force 
due  to  one  coil  pass  through  all  the  turns  of  the  other,  and  the 
value  of  M may  therefore  be  written  M LXLV  from  which 
it  is  at  once  seen  that  the  mutual-inductance  of  the  two  coils  must 
always  be  small  if  the  self-inductance  of  one  or  both  of  the  coils 
is  very  small,  while  the  mutual-inductance  may  be  large  if  both 
the  self-inductances  are  large.  It  is  shown  in  a preceding  par- 


452 


ALTERNATING  CURRENTS 


agrapli  that  the  value  of  M may  be  generally  written  Ms-n^Jt. 
In  this  expression  </q  is  dependent  on  nv  whence  it  is  shown 
that  My  nxnjc.  This  is  also  to  be  derived  from  the  relation 
M OC  v LxL2i  since  Lx  and  Z2  are  respectively  proportional  to  nx2 
and  n22.* 

The  summation  product  of  the  number  of  lines  of  force  which 
interlink  two  coils  by  the  numbers  of  turns  in  the  individual 
coils  will  not  change  by  changing  the  point  of  reference  from 
one  coil  to  the  other,  provided  the  reluctance  of  the  magnetic 
circuit  and  its  shape  are  unchanged,  and,  consequently,  the 
mutual-inductance  of  two  coils  is  the  same  when  measured  from 
either  coil.  When  the  magnetic  circuit  is  composed  wholly  or 
partly  of  iron,  it  is  necessary  to  have  the  same  distribution  of 
lines  of  force  in  the  parts  of  the  magnetic  circuit  when  the  two 
measurements  are  made,  in  order  that  they  may  give  equal 
results.  When  the  magnetic  circuit  is  made  up  partly  of  iron 
and  partly  of  non-magnetic  materials,  and  the  coils  are  dissim- 
ilar in  their  relations  to  the  magnetic  circuit,  as  in  the  case  of 
armature  and  field  windings  of  dynamos,  the  condition  of  equal 
distribution  of  lines  of  force  is  difficult  to  fulfill  on  account  of 
the  effects  caused  by  unequal  saturation  of  the  iron,  and  the 
value  of  M may  be  quite  different  when  measured  from  the  field 
winding  and  from  the  armature  winding.  This  difference  is  due 
to  the  difference  in  the  permeability  of  the  magnetic  circuit  and 
the  form  of  the  magnetic  paths,  during  the  two  measurements. 

As  shown  above,  the  mutual-inductance  is  homogeneous  with 
and  therefore  of  the  same  absolute  dimensions  as  self-inductance. 
Its  unit  is  therefore  the  Henry.  The  Henry  is  109  times  as  large 
as  the  G.G.S.  absolute  unit  of  inductance. f 

If  the  lines  of  force  due  to  one  coil  which  enter  another  do 
not  all  pass  through  all  the  turns  of  the  second  coil,  as  may  be 
tacitly  assumed  in  developing  the  foregoing  relations,  the  defi- 
nition of  mutual-inductance  still  holds  as  already  given,  but  the 
summation  of  the  linkages  of  lines  of  force  with  the  individual 
turns  must  be  taken  in  a manner  similar  to  the  summations 
required  to  obtain  the  self-inductance  of  a coil  through  all  the 
turns  of  which  all  of  the  self-induced  lines  of  force  do  not  pass.ij: 

If  a change  occurs  in  the  reluctance  of  the  magnetic  circuit 

* Art.  41. 


t Art.  42. 


1 Art.  43. 


MUTUAL  INDUCTION,  TRANSFORMERS 


453 


common  to  the  coils,  the  relative  positions  of  the  coils  remaining 
unchanged,  the  mutual  inductance  is  altered  from  M to  M'  = 

p 

ilf— , where  P and  P'  are  the  reluctances  of  the  path  before 

and  after  the  reluctance  is  changed,  such  as  by  inserting  an 

p 

iron  core.  The  ratio  — is  dependent  on  the  permeability  of 


the  iron  in  the  magnetic  circuit,  and  the  value  of  M must 
therefore  vary  with  the  current  in  the  coils  in  any  case  where 
iron  is  in  the  magnetic  circuit,  while  it  is  independent  of  the 
value  of  the  current  when  magnetic  material  is  absent. 

120.  The  Energy  of  Mutual  Induction.  — Assume  two  adjacent 
coils  with  a constant  mutual-inductance,  in  which  all  of  the 
flux  passes  through  all  the  turns  of  both  coils,  such  as  the 
superimposed  long  solenoids  described  in  Art.  119.  In  one  let 
a current  of  Ix  an^eres  flow,  and  in  the  other  a current  of  I2 
amperes,  the  two  magneto-motive  forces  being  additive  in  the 
mutual  magnetic  circuit.  The  number  of  linkages  of  lines  of 
force  due  to  the  first  coil  with  the  turns  of  the  second  is 
108JiJr  and  the  time  rate  of  change  in  this  number,  when  the 


current  in  the  first  coil  changes,  is 


10*Mdi 


dt 


h where  i1  is  the  in 


stantaneous  value  of  the  current  in  the  first  coil  while  it  is 
changing.  If  the  current  Ix  is  varied,  the  work  due  to 
mutual  induction  which  is  done  in  the  second  coil  in  chang- 
ing the  magnetic  field  against  the  effect  of  the  current  I2  is 
obtained  from  the  following  relations,  in  which  I2  represents  the 
current  in  the  secondary  coil  and  e2  represents  the  voltage  in- 


duced in  the  second  coil  by  the  time-rate  of  change 


of 


the  current  in  the  first  coil,  and  is  considered  positive  in  sign 
when  it  tends  to  increase  Z2  : 


and  div  = I2e2'dt ; * 

hence  dw  — — MI2div 


If  the  current  in  the  first  coil  is  changed  from  zero  to  ,Zj  (J2 
being  maintained  by  a battery  of  constant  voltage),  the  work 

* Art.  47. 


454 


ALTERNATING  CURRENTS 


done  on  the  second  coil  is  W=  — J MI2dix  = — MIXIV  which 
is  all  abstracted  from  the  electric  circuit  of  the  primary  coil 
and  stored  in  the  joint  magnetic  field.  If  the  current  Ix  falls 
again  to  zero,  this  work  is  restored  as  electrical  energy  to  the 
primary  circuit.  When  M varies  with  the  current,  the  work  is 
still  M1XIV  but  M in  the  expression  must  be  assigned  its  proper 
value  corresponding  to  the  maximum  magnetization  of  the  core. 
As  the  current  in  the  first  coil  rises  to  Iv  the  total  work  stored 
in  the  magnetic  field  is  evidently  the  sum  of  the  work  due  to 

A/2 

self  and  mutual  induction,  or  - -1  - MIXI2. 


If  the  current  is  caused  to  vary  at  the  same  time  by  a source 
of  energy  in  each  of  the  coils,  the  following  condition  exists  at 
any  instant.  The  instantaneous  voltage,  which  must  be  im- 
pressed on  each  coil  to  maintain  the  current  flow  therein,  is  the 


sum  of  the  active  voltage  (*i2),  the  reactive  voltage,  which 


is  equal  and  opposite  to  the  voltage  of  self-induction,  and  the 
reactive  voltage,  which  is  equal  and  opposite  to  the  voltage 
of  mutual-induction,  whence 


ei  — 0^2  + 


(1) 


e2  — -®2*2  d"  ^ 

where  ex  and  e2  are  the  instantaneous  impressed  voltages  in  the 
two  coils,  ix  and  i2  are  the  corresponding  instantaneous  currents 
in  the  two  coils,  Rv  R2,  Lv  Lv  are  the  resistances  and  self-in- 
ductances of  the  two  coils,  and  M is  their  mutual  inductance. 
By  transformation,  we  have 

(ex  — = Mdi2  + L1dil , (2) 

(e2  — i2R2~)dt  = Mdix  + L2div 

M,  Lv  and  L2  being  assumed  to  be  constant.  Multiplying 
these  equations  respectively  by  ix  and  i2  and  adding,  gives 

(ixex  4-  qe2)  dt  — (ixRx  + «22i?2)  dt 

= Lxixdix  + L2i2di2  + M(ixdi2  + i2di{). 


(3) 


MUTUAL  INDUCTION,  TRANSFORMERS 


455 


The  first  terra  of  the  left-hand  member  of  this  equation  rep- 
resents the  total  work  done  by  the  impressed  voltages  during 
the  interval  dt,  and  the  second  term  of  the  same  member  rep- 
resents the  work  expended  in  heat  during  the  same  interval  on 
account  of  the  currents  flowing  through  the  resistances  of  the 
coils.  The  difference  between  these  two  terms  represents  the 
work  done  on  the  magnetic  field,  and  this  is  represented  by  the 
right-hand  member  of  the  equation.  The  total  work  done  on 
the  magnetic  field  during  any  change  of  the  currents,  as  from 
zero  to  Ix  and  I2  amperes,  is  found  by  integrating  the  right- 
hand  member  of  the  equation.  Thus, 


J/, . Cr-  ■ • CIv‘-  ■ ■ L /2 

i\di\  + L<i iodi<i  + ( iydi 2 + i^di-y ) = ^ 

+ Ljl  + MIlIv  (4) 


This  is  all  absorbed  from  the  two  electric  circuits  and  stored 
in  the  magnetic  linkages,  which  are  set  up  by  the  rise  of  the 
currents.  If  the  currents  now  fall  to  zero  again,  the  same 
amount  of  energy  is  restored  to  the  electric  circuits  as  the  mag- 
netic field  is  restored  to  its  initial  condition.  The  work  MIXI2 
is  equally  divided  between  the  two  coils  when  Ix  = IT 

The  foregoing  relates  to  two  coils  receiving  their  current 
through  separate  electric  circuits,  but  if  the  two  coils  are  con- 
nected in  series  relation  in  one  electric  circuit,  so  that  they 
carry  the  same  current,  the  expression  for  the  work  stored  in 
the  magnetic  field  as  the  current  rises  from  zero  to  I amperes 
becomes 

M2  + M2+ilfj  2.  (5) 


When  the  two  coils  in  series  are  of  equal  self-inductances  and 
are  located  relatively  to  each  other  so  that  there  is  no  magnetic 
leakage,  in  which  case  Lx  — Z2  = M,  the  expression  represent- 
ing stored  work  becomes 

-I-2  + ^ + MI2  = 2 MJP  = 2 LP.  (6) 

2 2 J 

Equation  (3)  is  written  on  the  assumption  that  no  eddy  cur- 
rent or  hysteretic  losses  are  involved,  but  in  case  such  losses 
come  into  the  operation  an  additional  energy  term  representing 


456 


ALTERNATING  CURRENTS 


them  may  be  subtracted  from  the  first  term  in  the  left-hand 
member  of  the  equation.  Equation  (3)  is  also  written  on  the 
hypothesis  of  the  magnetic  fields  of  the  two  currents  being 
additive,  and  if  they  are  subtractive,  the  algebraic  sign  of  one 
of  the  currents  is  changed.  The  result  in  equation  (4)  then 


becomes 


MIX1 2; 


and  in  (5)  becomes 


while  (6)  reduces  to 


L1P+L.iP 
2 2 


- MP  = 0. 


121.  Transfer  of  Electricity  by  the  Effect  of  Mutual  Induction. 
Coils  in  Series  and  in  Parallel.  — Continuing  the  assumption 
of  two  coils  in  separate  electric  circuits  and  constant  L 
and  M,  suppose  that  no  voltage  is  initially  impressed  on  the 
second  coil  though  its  circuit  is  closed  ; then  when  the  current 
in  the  first  coil  is  changed,  the  conditions  in  the  second  coil 
are  given  from  the  equation 


Whence 

and 


+ -^2*2)  — 0- 


i2R  2dt  — — Mcli1  — L2di2 , 


di2. 


The  reason  for  the  two  zero  limits  of  the  third  term  of  the 
above  equation  is  evident  when  it  is  remembered  that  the  sec- 
ondary current  grows  from  zero  and  returns  to  zero  during  the 
operation.  Since  the  last  term  reduces  to  zero,  the  quantity 
of  electricity  which  is  transferred  in  the  second  coil  under  the 
inductive  influence  of  the  first  when  its  current  changes  from 
zero  to  7j  is 


MUTUAL  INDUCTION,  TRANSFORMERS 


457 


If  the  current  of  the  first  coil  is  now  brought  to  its  original 
value,  we  have 


The  two  quantities  are  equal  and  of  opposite  sign,  so  that  the 
transfer  of  electricity  in  the  secondary  coil  during  the  rise  and 
fall  of  the  primary  current  reduces  to  zero,  provided  the  origi- 
nal and  final  values  of  the  magnetism  of  the  magnetic  circuit 
are  the  same.*  If  the  current  in  the  first  coil  is  a simple  peri- 
odic one,  a periodic  current  of  the  same  frequency  is  set  up  in 
the  second  coil. 

The  instantaneous  voltages  in  the  two  coils  connected  in 
series  in  an  electric  circuit,  where  the  current  varies,  are 


r,  . t (^2  -jt/rdx 

ex  = Rxi  + — ± M-, 


= /fni  + Ln—r  i M' 
2 2 2 dt 


di 
dt ’ 


assuming  the  reluctance  of  the  magnetic  circuit  to  remain  con- 
stant; and  the  instantaneous  voltage  across  the  two  coils  is 

j; 

e = ( R j + R2)i  + ( Lx  + Z2  ± 2 M')  — • 


dx 

The  positive  or  negative  algebraic  sign  for  M - in  these  equa- 
tions must  be  selected  according  to  whether  the  coils  are  con- 
nected in  the  electric  circuit  so  as  to  cause  their  magnetic  effects 
to  aid  or  oppose  each  other  in  the  joint  magnetic  circuit.  Revers- 
ing the  connection  in  the  electric  circuit  of  the  winding  of  one  of 

the  coils  reverses  the  algebraic  sign  of  the  expression  related 
to  both  coils.  dt 

Upon  switching  a constant  voltage  E into  this  circuit,  the 
deficit  of  electricity  transferred  through  the  circuit  during  the 

]E 

rise  of  current  to  its  final  value  I = — — is 

Rx  + R2 


(Z,  +L,  ±2  31)1 

Rx  + R2 


coulombs. 


* Compare  Art.  46. 


458 


ALTERNATING  CURRENTS 


The  “ extra  current  ” occurring  at  switching  out  the  voltage 
without  changing  the  circuit  resistance  comprises  an  equal 
number  of  coulombs. 

Returning  now  to  the  equation 

e = (-Bj  + -B2)i  4"  + -^2^  2 T^)— ', 

assuming  e to  be  a sinusoidal  function  of  time,  so  that 
e — emsin  cot,  the  solution  of  the  equation  is,  after  a short  lapse 
of  time  when  the  current  wave  has  become  sinusoidal, 

i = --  sin  ( cot  — 9), 

z 

in  which  Z = V-B2  + [2  n rf(L1  + X2  ± 2 M )]2, 

where  R = Rx  4-  B2, 

and  Z = R +j  2 + i2  ± 2 3T), 

and  tan  9 = L^~  Ml . 

R 

In  case  Lx  = Zl2  and  there  is  no  magnetic  leakage  so  that 
M=  VX1X2  = X,  the  self-inductance  of  either  coil,  the  value  of 
Z becomes  either 

Z = V R2  4-  (8  irfL  )2,  or  Z = B, 

and  tan  6 = MU , 0r  tan  6=  0°. 

R 

If  the  coils  referred  to  in  the  two  preceding  paragraphs  are 
of  equal  self -inductances  and  are  located  relative  to  each  other 
so  that  there  is  no  magnetic  leakage,  it  is  obvious  from  the 
voltage  equation  that  (since  M then  equals  VX1X2  = X)  the 
reactive  voltage  caused  by  the  two  coils  in  series  is  either  four 
times  as  great  as  the  reactive  voltage  caused  bjr  one  of  the  coils 
alone,  uninfluenced  by  the  other,  or  is  zero  (in  the  first  case 
they  act  like  a single  coil  of  twice  the  turns  of  either);  and, 
also,  if  the  two  coils  are  of  equal  self-inductances,  but  are  lo- 
cated relative  to  each  other  so  that  M is  zero,  the  reactive 
voltage  caused  by  the  two  coils  is  twice  as  great  as  the  reactive 
voltage  caused  b}r  one  coil  alone. 

When  the  number  of  coils  in  series  is  «,  the  equations  of 


MUTUAL  INDUCTION,  TRANSFORMERS 


459 


voltage  become,  when  the  subscripts  1,  2;  1,  3;  etc.,  represent 
the  mutual  inductances  between  coils  of  those  numbers, 


ex  — Rxi  + Lx  — + ( ± Mx  2 ± Mx  3 ± • ■ 

rj'j 

e2  = ^2*  + -^2  ^ + ( ± ^2,  1 ± ^2,  3 i " 

en  = Rn%  + + ( ± j ± 2 ± • 

and  the  instantaneous  voltage  measured  across  the  series  of 
coils  is 

e = i1*R  + + 2 }'*  ± M. 

dt  dt 

The  impedance  of  such  a circuit  is 

Z = Ry+  [2  t rf(T[L  + 22^1,’"±ilf)]2, 

and  the  reactance  is 

X = 2 irf(XL  + 2 s 1)1 " ± i!f). 

As  far  as  the  effects  in  the  circuit  of  coils  in  series  are  con- 
cerned, the  mutual  inductance  between  the  coils  in  the  circuit 
merely  adds  to  or  deducts  from  the  reactions  caused  by  self- 
inductance. Mutual  inductance  between  coils  in  different 
electric  circuits,  however,  affords  different  results  by  causing  a 
transfer  of  energy  from  one  electric  circuit  to  the  other  when 
the  currents  vary  in  one  or  both  of  the  circuits. 

The  conditions  are  more  complex  when  the  mutually  induc- 
tive circuits  are  connected  in  parallel  to  the  same  main  circuit, 
instead  of  being  connected  in  series  as  heretofore  considered. 
In  case  of  two  mutually  inductive  coils  connected  in  parallel, 
the  voltage  formulas  are 

and  e = + (2) 

The  following  equations  may  be  derived  from  these  by  differ- 
entiation, 


460 


ALTERNATING  CURRENTS 


and 


de  r,  di , , t d2i,  7,rd2i0 

dt  = B'ft  + L'le±Mw’ 
de  = R„ih  + hfiz  ± MSt ' 


dt 


dt 


dt 2 


di2 


(3) 

(4) 


The  positive  sign  pertains  to  when  the  coils  are  con- 

nected to  the  electric  circuits  so  that  their  magnetic  effects  are 

di 

cumulative,  and  the  negative  sign  pertains  to  fff—  when  the 

dt 

magnetic  effects  of  the  coils  are  in  opposition. 

The  following  equations  in  i1  and  i2  are  obtained  by  multi- 
plying (1),  (3),  and  (4)  respectively  by  R2,  Z2,  and  T M,  and 
adding  to  get  (5),  and  multiplying  (2),  (3),  and  (4)  respec- 
tively by  Rv  t M,  and  Lv  and  adding  to  get  (6)  : 


R2e  + (Z2  T + (,R\L2  + L1R2)-^ 


d ^ 

dt 2’ 


Rxe  + (Zx  T M)  — — RxR2i2  + (ZjL2  + LXR^ 


+ (ZjZ2  — M2') 


d\ 
dt 2 


(5) 


(6) 


Equations  (5)  and  (6)  afford  exact  solutions  when  Rv  R2, 
Lv  L2,  and  M are  of  constant  values  and  the  voltage  is  a sine 

function  of  time.  In  this  case  e = em  sin  cot,  ^ = coem  cos  cut, 

dt 

and  — e-  = — (o2em  sin  cot.  Therefore  ^ ^ ^ = — to 2.  Substituting 
dt 2 dt 1 

these  values  in  equations  (5)  and  (6),  and  putting  RXR2  — 

o)2(Z1Z2  - M2)  = a,  RXL2  + LXR2  = b,  = Z)  and  = Z2 
gives  by  integration,* 

_ [Z^2  S^n  tut  + &)(Z2  T TZ)  COS  ^ q €-mii 

1 a + 6Z>  1 

■ _ r Ri  Sin  at  + <»(£,  T M)  cos  + ^ + ( g) 

a + hi) 


* Murray’s  Differential  Equations,  Chap.  VI. 


MUTUAL  INDUCTION,  TRANSFORMERS 


461 


The  exponential  terms  quickly  become  negligible  after  the 
current  is  started.  Neglecting  the  exponential  terms  and 
rationalizing  the  right-hand  member  of  each  of  these  equations 
by  multiplying  numerator  and  denominator  by  a — bD , simpli- 
fying the  expression,  and  factoring  the  resulting  coefficient 
in  the  numerator  gives 


. ^ emV[aR,+co2b(L,  t M)V+ co*[a(  L2t  M)- bE9J2  { . 

1 a2  -f  oo2b2  1 


= e”Vi?2 2 + a)2(^T.II2  Bi„(©t  - 0X), 

Va2  + co2b2 


(9) 


1 a 2 + c o2b2 


(10) 


= t-Vfil2  + a,2(Jl  sin («i  - A.)  ; 

V<!2  + •«* 

in  which 

tan  Qx=  ~ l1^  and  tan  d2  = *[<L\  ~ h^i\ 

1 aR2  + co2b  (i2  T df)  2 ai2x  -)-  &>25(f  x T df) 

It  follows  from  equations  (9)  and  (10)  that 


and 


j _ V R22  + 6)2 (A  t df )2  £ 
V a2  + o»2/i2 

j = a)2(f,  TdP)2  ^, 

Va2  + co2b2 

I , IRS  + co2(L,tM)2 

f2  V A*!2  + «2(fx  T J/)2 


7 


Va2  + o)2^2 


A V A22  + «2(  A2  t df)2 


zx  = 


+d 


^2  = - = 


VA22  + a)2(X2  t df)2  VA22  + ft)2(X2  T df)2' 
E 


Va2  + «2f>2 


h V^  + ^T#)2’ 


Z2  = 


:+d 


Vi?x2  + «2(Xx  T df  )2  V^!2  + ®2(£x  T df)2  ' 


462 


ALTERNATING  CURRENTS 


When  the  coils  are  superposed  so  that  there  is  no  magnetic 
leakage,  are  connected  in  the  electric  circuit  so  that  their 
magnetic  effects  are  cumulative,  and  the  resistances  and  self- 
inductances are  equal,  the  foregoing  expressions  reduce  to 

tan  dj  - tan  d2  — , 


Ji  = /2  = 


E 


Vi£2  + (2  coLy  ’ 


Z1  = Z2  = ^E 2+ (2  <oL) 2 ; 


since  the  named  conditions  make  Rx  = R2,  Lx  — L2  = M,  and 

di 

satisfy  the  uppermost  of  the  algebraic  signs  associated  with  M — 

and  It  will  be  observed  that  the  mutual  inductance  has, 
under  these  circumstances,  the  effect  of  doubling  the  reactance 
and  increasing  the  angle  of  lag,  compared  with  the  values 

(a> L and  tan1  for  corresponding  resistances  and  self-induct- 
ances with  M zero. 

It  is  also  to  be  observed  from  the  equations  that  when  the 
coils  are  connected  in  the  electric  circuit  so  as  to  make  their 
magnetic  effects  cumulative,  the  effect  of  the  mutual  inductance 
is  always  to  increase  the  impedance  of  the  coils.  The  impedance 
is  a maximum  when  M — V LXL2 ; that  is,  when  magnetic  leakage 
is  negligible. 

When  the  coils  are  superposed  as  before,  but  connected  in 
the  electric  circuit  so  that  their  magnetic  effects  are  in  opposi- 
tion, Rx  being  equal  to  R2  and  Lv  Lv  and  M being  equal  to 
each  other,  the  equations  reduce  to 

tan  6X  — tan  02  = 0, 

I — I — — 
i2~ir 


Zx  = Z2  = R. 

The  last  condition  satisfies  the  lower  one  of  the  algebraic  signs 
di 

associated  with  M — and  M.  The  mutual  inductance  under 
• dt 

these  circumstances  neutralizes  the  effects  of  self-inductance 
and  reduces  the  reactance  of  the  coils  to  zero. 


MUTUAL  INDUCTION,  TRANSFORMERS 


463 


When  the  magnetic  effects  of  the  coils  are  in  opposition,  the 
effect  of  their  mutual  inductance  is  always  to  reduce  the  im- 
pedance and  angle  of  lag.  The  maximum  influence  of  the 
mutual  inductance  occurs  when  M = V LXL2,  that  is,  when  there 
is  no  magnetic  leakage. 

The  formulas  representing  the  reactions  of  coils  in  parallel 
may  be  extended  to  any  number  of  coils  larger  than  two,  by 
the  same  processes  as  are  exhibited  in  the  foregoing.  If  either 
branch  circuit  has  electrostatic  capacity  in  series  with  its  resist- 
ance and  self-inductance,  a term  representing  condenser  voltage 
must  be  added  to  the  voltage  equations  corresponding  to  equa- 
tions (1)  and  (2).  When  the  capacity  measured  from  one  coil 
to  another  is  appreciable,  it  may  be  treated  approximately  by 
considering  it  as  a condenser  composing  a separate  branch  in 
parallel  with  the  coils ; but  an  exact  solution  of  this  case  requires 
the  use  of  equations  which  represent  the  effects  of  distributed 
resistance,  inductance,  and  capacity,  and  have  the  character  of 
those  developed  in  a later  chapter. 

122.  Ratio  of  Transformation  in  a Transformer. — The  formulas 
of  Art.  119  show  that  the  voltage  developed  in  a secondary 
coil  is  at  any  instant 

, _ d(Mi^) 

2 “ dt  • 


If  the  current  wave  is  a sinusoid,  this  becomes 


eJ  = 


d ( Milm  sin  cot ) 
dt 


If  the  conditions  require  that  M be  treated  as  a variable 
dependent  upon  the  varying  permeability  of  an  iron  core,  this 
equation  is  practically  insolvable.  For  a close  approximation 
to  practical  conditions,  however,  it  is  sufficient  to  assume  M as 
having  a constant  value  which  depends  upon  the  iron  of  the 
core  and  the  maximum  magnetic  density  used  in  the  trans- 
former. The  equation  then  becomes  e2  = — • The 


maximum  value  of  the  voltage  is  then  e2m'  = 2 7rfMilm , where  / 
is  the  frequency  of  the  current  wave,  since  da/dt  = w.  The 
effective  value  of  the  secondary  voltage  is  therefore  evidently 
E2  = — 2 7i fMIv  where  Ix  is  the  effective  primary  current.  If 


ALTERNATING  CURRENTS 


464 

the  secondary  circuit  is  open,  the  following  equations  may  be 
written 

j- 

1 VE*  + 4 7 T2f*Lf 

while  — E[  = 2 7 t/LxIx  = ~ i where  — E[  is  numeri- 
cally equal  but  opposite  to  the  total  induced  primary  counter- 
voltage.  (This  includes  the  voltage  induced  by  both  the  leak- 
age and  joint  magnetic  flux.)  Ex  has  been  used  to  represent 
the  impressed  voltage,  and  the  maximum  number  of  lines  of 
force  in  the  cycle.  If  the  resistance  of  the  primary  winding  is 
considered  negligible,  the  former  equation  becomes 


and 


2 t rfL1 

Ex  = 2 t TfLxIx  =-E[; 


and,  in  this  case,  if  the  primary  and  secondary  coils  are  so  com- 
pletely superposed  that  there  is  no  magnetic  leakage,  the  value  of  M 
becomes  M — Vi1Z2  ; whence 


E^  _ 2 irfLy  Ix  __  Lx  or  Ex  _ VXj~  _ 
2 7 TfMIx  ^L~L2  Vi; 


En 


But  — 


L,  since  the  magnetic  circuit  of  the  two  coils  is 


assumed  to  be  identical,  that  is,  M = V LXLV  and  consequently 
there  is  no  magnetic  leakage.  Therefore 

o o 


Ex 


= s. 


n 


2 


In  other  words,  if  the  active  voltage  in  the  primary  circuit  may 
be  considered  negligible  when  compared  with  the  impressed 
voltage,  and  there  is  no  leakage  of  magnetic  lines,  the  ratio  of 
the  impressed  voltage  to  the  voltage  induced  in  the  secondary 
winding  is  equal  to  the  ratio  of  the  number  of  turns  of  wire  in 
the  two  coils.  The  ratio  of  the  primary  voltage  and  the  secon- 
dary voltage  of  a transformer  is  commonly  called  the  Ratio  of 
transformation.  The  ratio  of  transformation  of  well-designed 
transformers  intended  for  use  on  constant  voltage  circuits  is 


MUTUAL  INDUCTION,  TRANSFORMERS 


465 


practically  equal  to  — when  the  secondary  circuit  is  open, 
n2 

showing  that  the  assumption  that  the  active  voltage  and 
magnetic  leakage  are  negligible  in  most  commercial  trans- 
formers, when  the  secondary  circuit  is  open,  is  entirely  allow- 
able. An  example  will  show  this  in  a striking  manner.  In  a 
certain  transformer  of  22.5  kilowatts  capacity  the  electrical  re- 
sistance of  the  primary  winding  is  practically  1 ohm  and 
the  inductance  is  9.1  henry s.  At  a frequency  of  60  cycles 
per  second  and  a voltage  of  2000  volts,  the  square  of  the  value 
of  the  reactance  is  nearly  12,000,000.  In  another  transformer 
of  11.25  kilowatts  capacity,  designed  for  2400  volts  primary 
voltage,  the  value  of  the  electrical  resistance  of  the  primary 
winding  is  6.45  ohms  (when  squared  41.6),  and  the  square 
of  the  value  of  the  reactance  is  10,000,000.  In  three  other 
transformers  designed  for  a voltage  of  1000  volts  and  respec- 
tively of  7.5,  4.5,  and  1.5  kilowatts  capacity,  the  electrical  re- 
sistances of  the  primary  windings  are  1.16,  2.15,  and  8.90  ohms, 
while,  at  a frequency  of  60  cycles  per  second,  the  squares 
of  the  values  of  the  reactances  are  respectively  25,000,000, 

31.000. 000,  and  100,000,000;  and  in  a transformer  of  .5  kilo- 
watt capacity,  the  electrical  resistance  of  the  primary  winding 
is  25  ohms  and  the  square  of  the  value  of  the  reactance  is 

400.000. 000.  In  each  of  these  cases,  which  represent  common 
practice  in  the  construction  of  transformers,  the  value  of  Rx2 
is  entirely  negligible  when  compared  with  4 7 

When  the  secondary  circuit  of  a transformer  without  magnetic 
leakage  is  closed,  neither  Rx  nor  i?2  can  be  neglected,*  and  then 
the  ratio  of  transformation  evidently  is  decreased  when  the 
secondary  voltage  is  higher  than  the  primary,  and  is  increased 
when  the  secondary  voltage  is  lower  than  the  primary  on  account 
of  voltage  due  to  the  current  flowing  through  Rx  and  R2  (that 
is,  for  a given  impressed  primary  voltage  the  secondary  terminal 
voltage  is  decreased). 

123.  Magnetic  Leakage  in  Transformers.  — The  primary  and 
secondary  coils  in  the  discussion  of  Art.  122  are  supposed  to  be 
so  sandwiched  together  that  magnetic  leakage  is  negligible 
when  there  is  no  current  in  the  secondary  coil.  This  is  not 
necessarily  the  case.  A case  when  magnetic  leakage  is  always 

* Art.  127. 

2 H 


466 


ALTERNATING  CURRENTS 


present  is  shown  in  Fig.  267.  From  the  figure  it  is  evident 
that  even  if  no  current  flows  in  the  secondary  winding,  the 
counter-voltage  in  the  primary  winding  will  be  greater  per  turn 
of  wire  than  the  voltage  induced  per  turn  in  the  secondary 
winding  ; hence,  even  if  the  self-induced  or  counter-voltage  in 
the  primary  winding  is  practically  equal  and  opposite  to  the 
impressed  voltage,  the  ratio  of  transformation  will  be  changed 
by  this  relative  decrease  of  the  secondary  induced  voltage.  If 
a current  flows  in  the  secondary  winding,  the  self-induction 
due  to  magnetic  leakage  in  the  secondary  (lines  of  force  link- 
ing with  the  secondary  coil,  but  not  linking  with  the  primary 
coil)  will  still  further  reduce  the  active  secondary  voltage,  and 


Fig.  267.  — Diagram  for  showing  Transformer  Leakage,  a,  a,  leakage  flux; 
b,  b,  mutual  flux. 


the  ratio  of  transformation  will  be  further  changed.  The  effect 
of  magnetic  leakage  in  altering  the  ratio  of  transformation 
(decreasing  the  proportional  voltage  induced  in  the  secondary 
winding  by  decreasing  the  magnetic  flux  passing  through  it) 
was  early  shown  by  an  experiment  reported  by  Professor 
Ryan.*  In  the  experiment  recorded  by  him,  the  primary  and 
secondary  coils  were  wound  on  opposite  sides  of  a laminated 
iron  ring  with  much  the  same  arrangement  as  indicated  in 
Fig.  267.  The  number  of  turns  in  the  primary  and  secondary 

windings  were  respectively  500  and  155,  or  -A  = 3.2.  When  a 

n2 

voltage  of  75.6  volts  was  impressed  upon  the  primary  winding 
with  the  secondary  circuit  open,  a voltage  of  only  16.4  volts 

* Some  experiments  upon  Alternating  Current  Apparatus,  Trans.  Amer. 
Inst.  E-  E.,  Vol.  7,  p.  324. 


MUTUAL  INDUCTION,  TRANSFORMERS 


467 


was  induced  in  the  secondary  winding,  or  — i = 4.6.  The 

-<8 

whole  difference  in  the  two  ratios  was  due  to  magnetic  leakage, 
and  the  magnitude  of  the  difference  shows  that  M was  much  less 
than  VXjig.  The  magnetic  leakage  was,  in  fact,  nearly  30 
per  cent ; that  is,  the  number  of  lines  of  force  that  linked  with 
the  primary  winding,  but  not  with  the  secondary  winding,  was 
30  per  cent  of  the  total  magnetic  flux  set  up  in  the  magnetic 
circuit. 

The  change  in  the  ratio  of  transformation,  just  spoken  of, 
occurred  when  the  secondary  circuit  was  open.  When  current 
is  permitted  to  flow  in  the  secondary  circuit  of  such  an  arrange- 
ment, the  effect  is  much  more  striking,  for  under  such  condi- 
tions the  magneto-motive  force  of  the  secondary  winding  op- 
poses that  of  the  primary  winding,  and  there  is  a strong  tendency 
to  force  magnetic  flux  across  the  air  space  surrounding  the  coils. 
That  is,  because  of  the  counter  magneto-motive  force  of  the  sec- 
ondary winding,  the  difference  of  magnetic  potential  between  the 
points  A and  B in  Fig.  267  may  be  very  great.  When  the  sec- 
ondary current  is  zero,  the  maximum  magneto-motive  force 
tending  to  send  flux  from  A to  B through  leakage  paths  is 
measured  by  m = <&P,  where  <I>  is  the  maximum  flux  in  the 
core  and  P is  the  reluctance  in  the  iron  circuit  connecting  those 
points  through  the  secondary  coil.  When  a current  flows  in 
the  secondary  winding,  the  maximum  magneto-motive  force  be- 


tween A and  B is  m — <t> P + 


V2  x 4 7t«2T2 
10 


where  the  second  term 


on  the  right  represents  the  part  of  the  primary  magneto-motive 
force  equal  and  opposite  to  the  back  or  counter  magneto- 
motive force  of  the  secondary  winding,  which  in  commercial 
transformers,  except  for  small  secondary  loads,  is  many  times 
larger  than  the  first  term.  The  values  of  <E>  and  P usually 
change  slightly  when  current  flows  in  the  secondary  coil. 

The  effect  of  magnetic  leakage  on  the  action  of  a transformer  is 
analogous  to  the  effect  which  would  be  produced  on  a transformer 
without  leakage  by  inserting  coils  having  inductive  reactance, 
i.e.  Reactance  or  Impedance  coils,  in  the  primary  and  secondary 
circuits  outside  of  the  transformer,  as  is  illustrated  in  Fig.  268. 
These  coils  would  have  such  self-inductances  as  to  increase  the 
self-inductances  of  the  primary  and  secondary  circuits  in  the 


468 


ALTERNATING  CURRENTS 


ratio  of  a : 100,  where  a is  the  magnetic  leakage  in  per  cent. 
Since  leakage  causes  a proportional  increase  in  the  apparent  self- 
inductance of  the  primary  and  secondary  circuits,  it  causes  an 
equivalent  lag  of  the  currents  in  the  two  circuits.  The  react- 
ance of  the  windings  of  a transformer  which  is  caused  by  the 
leakage  flux  may,  for  convenience,  be  called  the  Leakage 
reactance,  to  distinguish  it  from  the  reactance  caused  by  the 
total  of  magnetic  linkages.  The  part  of  the  self-inductance, 
which  is  due  to  the  leakage  flux  may  also,  for  convenience,  be 
called  the  Leakage  inductance,  to  distinguish  it  from  the  total 
self-inductance  of  the  windings. 


TRANSFORMER 


The  effect  of  magnetic  leakage  in  transformers  which  trans- 
form at  constant  primary  voltage  — Constant  voltage  trans- 
formers — is  evidently  to  interfere  with  the  regulation  of  the 
secondary  voltage,  so  that  in  such  transformers  the  leakage 
paths  are  made  of  as  high  reluctance  as  possible.  While  Con- 
stant current  transformers,  that  is,  those  which  transform  from 
constant  voltage  in  the  primary  circuit  to  constant  current  in 
the  secondary  circuit,  depend  upon  leakage  for  regulation.*  In- 
duction motors  are  affected  in  much  the  same  way  as  constant 
potential  transformers,  with  which,  in  principle,  they  have  much 
in  common. f 

124.  Diagram  of  Transformer  with  Magnetic  Leakage. 
Equivalent  Circuit. — The  effect  of  magnetic  leakage,  as  dis- 
cussed in  the  preceding  article,  may  be  shown  bj"  a simple  dia- 
gram, such  as  Fig.  269.  In  this  diagram,  0 C represents  the  vector 
magneto-motive  force  caused  by  the  current  Iv  which  flows  in 
the  secondary  circuit  when  voltage  E1  ( OE)  is  impressed  on  the 


* Art.  143. 


t Art.  192. 


MUTUAL  INDUCTION,  TRANSFORMERS 


469 


primary  winding,  as  in  Fig.  266,  and  this  magneto -motive 
force  is  just  balanced  by  the  magneto-motive  force  produced 


Fig.  269.  — Diagram  showing  the  Relations  of  Currents  and  Voltages  in  a Trans- 
former with  Large  Magnetic  Leakage. 


by  the  component  1^  ( 01  of  the  diagram)  of  the  primary  cur- 
rent. These  magneto-motive  forces  are  equal  and  opposite  to 


470 


ALTERNATING  CURRENTS 


each  other  as  far  as  the  joint  magnetic  path  through  the  pri- 
mary and  secondary  coils  are  concerned,  the  mutual  flux  being 
produced  by  the  magneto-motive  force  of  primary  component 
OA ; but  the  magneto-motive  forces  of  I2  and  1\  act  in  parallel 
on  the  paths  of  leakage  flux,  such  as  the  leakage  path  from  A to 
B in  Fig.  267,  and  set  up  leakage  fluxes  which  create  the  re- 
actance-induced voltages  U1Li  and  H2I/2  (OH  and  OG  of  the  dia- 
gram). Except  for  the  effect  of  the  magnetizing  current  1^  on 
these  reactance-induced  voltages  are  opposite  to  each  other. 
They  lag  substantially  90°  behind  their  respective  inducing 
magneto-motive  forces,  since  the  reluctance  of  each  leakage  path 
is  mostly  outside  of  the  magnetic  material  and  there  is  therefore 
no  appreciable  iron  loss  angle  of  advance  of  the  current. 

For  conqflete  accuracy,  the  voltage  induced  by  the  primary 
leakage  flux  should  be  taken  in  quadrature  with  the  total  pri- 
mary current,  OB , and  it  is  not  exactly  opposite  to  the  voltage 
induced  by  secondary  leakage  flux.  It  may  be  considered  as 
comprising  two  components,  of  which  one  is  OH  in  lagging 
quadrature  with  01  and  due  to  the  leakage  flux  caused  by  the 
01  component  of  primary  current,  and  the  other  is  in  lagging 
quadrature  with  OA  and  is  due  to  the  leakage  flux  caused  by 
the  exciting  component  OA  of  the  primary  current.  The  first 
of  these  components  is  opposite  to  OG.  For  simplicity,  the 
second  of  the  components  is  neglected  in  the  figure,  since  it  is 
very  small  in  commercial  transformers  because  the  exciting 
current  is  small.  If  the  highest  accuracy  is  desired,  both  com- 
ponents may  be  introduced  in  the  diagram  and  treated  in  a 
manner  corresponding  to  the  treatment  of  OH  in  the  following 
discussion.  The  reactance-induced  voltages  OH  and  OG  are 
not  necessarily  of  equal  numerical  values  per  turn  in  the  wind- 
ings, because  the  reluctance  of  the  leakage  paths  about  the 
primary  and  secondary  coils  may  not  be  the  same. 

The  diagram,  Fig.  269,  is  drawn  with  all  vectors  reduced  to 
primary  equivalents  by  multiplying  secondary  voltages  and 
dividing  secondary  currents  by  the  ratio  of  the  turns  in  the 
primary  and  secondary  windings.  The  counter-induced  volt- 
age in  the  primary  winding  is  the  vector  sum  of  E'lm,  the  voltage 
induced  by  the  mutual  flux  threading  both  the  primary  and 
secondary  windings,  and  represented  by  OF  in  the  diagram, 
and  FUl,  the  voltage  induced  by  the  primary  leakage  flux  and 


MUTUAL  INDUCTION,  TRANSFORMERS 


471 


represented  by  OH  in  the  diagram.  The  secondary  induced 
voltage  E2m  is  represented  by  OF  in  the  diagram.  The  voltage 
drops  in  the  secondary  circuit  when  current  flows  therein  are: 
OJ  = NT. \ which  is  the  drop  of  voltage  E2R2  in  the  resistance  of 
the  secondary  winding  when  current  00  flows;  TF=  — OGr , 
which  is  the  drop  of  voltage  E2Lt  required  to  overcome  the 
secondary  leakage  reactance  ; and  ON  which  is  the  drop  of 
voltage  through  the  load,  i.e.  is  the  terminal  voltage  E2. 

The  voltage  drops  in  the  primary  circuit  are:  0L=  — OF , 
which  is  the  component  —Elm  of  impressed  voltage  required  to 
overcome  the  counter  voltage  set  up  in  the  primary  winding  by 
the  mutual  flux  which  threads  botli  windings  ; LM  4-  PE= 
OK  + OR  = 0 V \ which  is  the  drop  in  the  resistance  of  the  pri- 
mary winding  caused  by  the  flow  of  primary  current  OB  ; 
MP  = — OH , which  is  the  component  of  impressed  voltage 
— E1Ll,  required  to  overcome  the  primary  leakage  reactance. 
The  impressed  voltage  is  OE—  OL  + LM+MP  + PE.  The 
primary  angle  of  lag  is  dv  The  angle  between  the  secondary 
current  and  secondary  induced  voltage  is  angle  TOF,  but  the 
angle  of  lag  of  the  secondary  current  with  respect  to  the  ter- 
minal voltage  is  zero  in  the  diagram;  that  is,  the  external  sec- 
ondary circuit  is  assumed  to  be  non-reactive.  The  vector  OB' 
is  the  total  voltage  representing  power  expended  in  the  primary 
and  secondary  circuits,  and  HE  is  the  total  reactive  voltage  in 
primary  and  secondary  circuits. 

The  active  secondary  voltage  0 T is  E2Rn  + N2R  — E2(R  .+«) 
which  is  required  to  send  the  secondary  current  I2  through  the 
given  resistance  of  the  secondary  coil  and  external  circuit.  The 
value  of  E2(Ri+R)  is  determined  from  the  vector  diagram  com- 
prising itself  and  the  induced  voltages  E^  and  E2L^  the  relation 
being  E2(R 2+  R)  = E2m  — E2Ls.  The  voltage  used  in  the  secondary 
coil  resistance  is  E2R,  and  E2R  is  the  drop  in  the  load.  The  pri- 
mary winding,  being  cut  by  the  same  flux  that  sets  up  E2m , has 
a counter-voltage  Elm  set  up  in  it,  in  phase  with  and  equal  to 
sE2m , where  s is  the  ratio  of  the  number  of  primary  turns  to 
secondary  turns.  The  impressed  primary  voltage  E1  must  have 
a component  equal  and  opposite  to  this  Elm.  The  impressed  vol- 
tage Ex  must  also  furnish  the  voltage  E1R  required  to  drive  the 
element  of  the  primary  current  1 which  lias  a magneto-motive 
force  equal  and  opposite  to  that  of  I2,  through  the  primary  coil 


472 


ALTERNATING  CURRENTS 


resistance.  It  must  also  have  a component  — ElLi  equal  and  op- 
posite to  the  primary  leakage  voltage  ElL.  These  three  vol- 
tages are  closed,  as  indicated  in  Fig.  269,  by  the  vector  voltage 
represented  by  OP , forming  the  vector  polygon  OL3IPO.  But 
i^,the  exciting  current,  joins  vectorially  with  Ix,  in  making  up 
the  total  primary  current  Iv  and  I is  in  advance  of  Elm  by 
an  angle  of  90°  plus  the  iron  loss  angle  of  advance.  Its  com- 
ponent OQ  is  in  quadrature  with  OF  and  the  angle  AOQ  is  the 
iron  loss  angle  of  advance.  The  voltage  required  in  phase 
with  7m  to  drive  it  through  the  resistance  of  the  primary  circuit 
is  Ec  (=  OR),  and  this  must  therefore  be  vectorially  added  to 
the  voltage  represented  by  OP.  The  primary  voltage  drops 
from  the  vector  polygon  OLMPEO , which  has  the  resultant 
OE  equal  to  Ev  Instead  of  separating  the  voltage  drop  due 
to  the  resistance  of  the  primary  coil  into  two  components  Exr^ 
and  Ec , its  combined  value  in  phase  with  Ix,  the  total  primary 
current,  might  have  been  used  as  was  done  in  Fig.  266. 
The  primary  and  secondary  phase  angles  are  6X  and  02, 
the  latter  in  this  case  being  zero  as  the  load  assumed  is  non- 
reactive. 

The  triangle  ODP  is  especially  worthy  of  notice.  The  side 
OB  represents  the  voltage  that  would  be  necessary  to  drive 
7/  through  the  resistance  of  the  primary  coil  Rx  plus  the 
voltage  necessary  to  drive  the  secondary  current  72  divided 
by  s through  the  resistances  of  the  secondary  coil  R2  and 
external  load  resistance  P multiplied  by  s2.  Dividing  72 
by  s gives  a quotient  equal  to  7/,  and  therefore  71'2 

(R0  + R)s2  = I22CR2  -f-  R).  For  this  reason  2 . R2s2,  Rs2,  and 

s 

E2ms  are  sometimes  called  the  primary  equivalents  of  the 
secondary  quantities.  For  like  reasons  secondary  reactances 
and  impedances  must  be  multiplied  by  s2  to  reduce  to  pri- 


mary equivalents;  thus,  = V (_R2 + R)2 + (x2 + x)2, 

72  s2Ix 

where  x2  and  x are  the  reactances  of  the  secondary  coil  and  load 
respectively,  or  s2  V(if2  + R)2- f (a-2  + x)2.  The  side  OB 


of  the  triangle  OBP  is  thus  an  active  component  of  the  voltage 
represented  by  OP,  while  the  side  BP,  90°  therefrom,  is  reactive 


MUTUAL  INDUCTION,  TRANSFORMERS 


473 


and  equal  to  the  primary  leakage  voltage  drop  ElL  plus  the 
equivalent  secondary  leakage  voltage  drop  sE2L^. 

Such  a triangle  of  voltages  can  also  be  formed  if  the  secondary 
winding  is  replaced  by  a similar  coil  having  s times  as  many 
turns  and  a resistance  equal  to  s2i?2,  connected  in  series  with 
and  magnetically  opposed  to  the  primary  coil  and  in  series  with 
a load  circuit  in  which  the  resistance  equals  s2R.  Such  an  ar- 
rangement is  diagramed  in  Fig.  270.  Then  OP  (Fig.  269)  is  the 


Fig.  270.  — A Conventional  Arrangement  of  Coils  to  absorb  the  Same  Power  and 
produce  the  Same  Reactances  as  a Transformer  of  Equivalent  Impedance. 

voltage  measured  across  the  three  parts,  i.e.  the  first  coil,  the 
second  coil,  and  the  load;  OP  is  the  active  voltage,  and  PP  is 
the  leakage  reactive  voltage,  as  in  the  case  of  the  transformer 
discussed  above.  Under  this  arrangement  there  is  no  mutual 
flux  in  the  core,  but  only  the  leakage  flux.  To  make  the  sub- 
stitution complete,  a current  equal  in  scalar  value  and  phase  to 
1^  must  be  shunted  around  the  second  coil  and  the  external 
load.  The  magneto-motive  force  of  this  current,  when  it  flows 
through  the  first  coil  to  the  dividing  point,  sends  the  same 
magnetic  flux  through  the  core  as  1^  sets  up  in  the  transformer, 
and  by  which  the  voltages  Elm  and  E 2m  are  induced  (Fig.  269). 
This  current  also  combines  with  the  impressed  voltage  to  fur- 
nish the  power  absorbed  by  the  iron  losses.  The  component  of 
the  impressed  voltage  required  to  drive  1^  through  the  primary 
resistance  is  then  Ec;  this  combined  vectorially  with  the  voltage 


474 


ALTERNATING  CURRENTS 


represented  by  OP  equals  the  impressed  voltage  Ev  which  is 

required  to  make  the  currents  Ix  and  - I2  flow  respectively 

through  the  first  and  second  coils,  analogously  to  the  trans- 
former. The  counter-voltages  set  up  in  the  two  coils  by  the 
mutual  flux  which  threads  both  coils  exactly  counterbalance 
each  other,  and  have  no  effect  on  either  the  voltage  or  current 
of  the  supply  wires  or  the  load,  so  for  purposes  of  analogy 
they  may  be  considered  to  be  absent.  The'  combination  of 
resistance  and  reactance  necessary  to  shunt  the  current  1^ 
around  the  second  coil  and  load  is  indicated  in  Fig.  270  by 
the  part  marked  I Regulator. 

In  this  manner  two  coils  in  series  opposition  have  been 
substituted  for  the  transformer  reduced  to  terms  of  unity 
transformation,  in  which  the  equivalent  voltages  for  the 
transformer  are  impressed  on  the  system,  driving  equivalent 
currents  at  the  same  angles  of  lag,  and  expending  the  same 
power  in  each  coil  and  in  the  load. 

If  the  secondary  load  of  a transformer  is  reactive,  an  inspec- 
tion of  Fig.  269  will  show  that  02  and  9X  tend  to  increase,  while 
with  a load  containing  electrostatic  capacity  these  angles  tend 
to  lessen,  or  to  reverse  in  sign  if  the  capacity  is  sufficiently  large.* 

The  leakage  fluxes  and  internal  losses  of  voltage  indicated  by 
Fig.  269  are,  for  the  purpose  of  obtaining  a clear  diagram, 
made  much  greater  in  proportion  to  the  impressed  and  mutual 
induced  voltages  than  would  be  found  in  commercial  constant 
voltage  transformers.  However,  the  leakage  fluxes  used  for 
the  diagram  are  not  necessarily  excessive  for  conditions  in  in- 
duction motors  and  constant  current  transformers,  to  which  the 
diagram  is  also  applicable.  Likewise,  the  exciting  current  1 4 
has  been  given  a large  value  in  Fig.  269,  for  the  same  reasons. 
It  is  usually  so  small  in  constant  voltage  transformers  that  the 
voltage  drop  OP  may  be  considered  numerically  equal  to  and  of 
the  same  phase  as  OE  (Fig.  269)  for  the  solution  of  such  prob- 
lems as  usually  arise  in  commercial  circumstances.  However, 
when  the  load  current  considered  is  a small  fraction  of  the  full 
load  current  or  the  transformer  is  special,  so  that  in  either  case 
the  drop  due  to  the  exciting  current  is  material,  the  exact  tri- 
angle representing  apparent  primary  impedances  and  resultant 

* Art.  131. 


MUTUAL  INDUCTION,  TRANSFORMERS 


475 


active  and  reactive  voltages  is  OD'  E.  In  this  triangle,  when 
scaled  as  a triangle  of  voltages,  the  side  in  quadrature  to  the 
current  Ix  = OB  must  be  composed  of  the  quadrature  projection 
of  the  drops  of  primary  voltage  that  are  due  to  those  induced 
in  .the  primary  and  secondary  coils  and  load  and  is  equal  to 
D' E.  It  will  be  here  remembered  that  OH , when  fully  ex- 
pressed, is  at  right  angles  to  OB.  The  active  voltage  OD'  in 
phase  with  Ix  must  be  composed  of  the  projection  of  the  active 
drops  of  voltage  caused  by  the  work  done  in  driving  the 
current  through  the  primary  and  secondary  coils  and  the  load. 
When  the  triangle  OD' E is  scaled  to  represent  impedances, 
i.e.  voltages  divided  by  currents,  the  line  OE  is  the  total  ap- 
parent impedance,  0D\  the  apparent  resistance,  and  D'E,  the 
apparent  reactance  offered  by  the  two  windings  and  the  load 
of  the  transformer.  The  power  input  is  of  course  EXIX 
cos  0X  — OE  x OB  x cos  Z.EOD'  and  the  output  E2I2  cos  02  = 
00  x ON. 

125.  Transformer  Exciting  Current.  — In  the  case  of  an  ideal 
transformer,  that  is,  one  without  losses  or  magnetic  leakage, 
the  lag  of  the  primary  current,  when  the  secondary  circuit  is 
open,  is  90°  with  respect  to  the  impressed  voltage  ; for  Rx  and 
cos  9X  are  assumed  to  be  zero.  Since  the  induced  secondary 
voltage  lags  behind  the  magnetism,  which  is  in  phase  with  the 
primary  current  when  the  secondary  circuit  is  open,  by  an 
angle  of  90°,  the  phases  of  the  primary  impressed  voltage  and 
the  secondary  induced  voltage  are  exactly  180°  apart ; that  is, 
they  are  in  exact  opposition.  The  current  in  the  primary  circuit 
of  an  ideal  transformer,  when  the  secondary  is  open,  is  all 
reactive  (i.e.  wattless),  and  of  a magnitude  which  depends 
only  upon  the  total  inductance  L ' of  the  primary  coil. 

On  the  other  hand,  the  losses  due  to  hysteresis  and  eddy 
currents  in  the  iron  core  and  to  resistance  in  the  primary  coil  are 
by  no  means  negligible  in  commercial  transformers,  but  are  of 
such  a magnitude  as  to  decrease  the  lag  of  the  primary  current 
until  the  power  factor  of  the  primary  circuit  is  ordinarily 
between  50  per  cent  and  70  per  cent  when  there  is  no  current 
in  the  secondary  circuit  ; but  the  magnetism  in  the  core 
remains  nearly  in  phase  with  the  reactive  component  of  the 
primary  current  and  is  almost  90°  behind  the  phase  of  the 
primary  voltage,  so  that  the  primary  impressed  and  secondary 


476 


ALTERNATING  CURRENTS 


induced  voltages  are  still  almost  in  opposition.  The  current 
which  flows  in  the  primary  circuit  when  the  secondary  circuit 
is  open  may  therefore  be  considered  as  composed  of  two  compo- 
nents, one  of  which  supplies  the  power  required  to  make  up 
the  transformer  losses,  and  the  other  of  which  serves  simply 
for  setting  up  the  magnetism,  and  is  therefore  wattless.  The 
primary  current  which  flows  when  the  secondary  circuit  is 
open  is  sometimes  called  the  leakage  current,  open  circuit  cur- 
rent, or  magnetizing  current.  In  this  volume,  however,  the 
more  satisfactory  term,  Exciting  current,  is  used,  the  term  mag- 
netizing current  being  reserved  to  apply  only  to  the  magnet- 
izing (i.e.  reactive)  component  of  the  exciting  current. 

It  is  desirable  to  look  a little  farther  into  the  effect  upon  the 
cyclic  curve  of  iron  loss  which  is  produced  by  substituting  a 
sinusoidal  exciting  current  for  the  irregular  current  which  actu- 
ally flows,  as  has  been  done  in  the  transformer  diagram  of  Figs. 
265,  266,  and  269,  and  others  following.  In  Fig.  271,  <M>  is  a 


Fig.  271.  — Conventional  Cyclic  Curve  of  Iron  Loss,  constructed  from  the  Curve 
of  Flux  in  a Transformer  Core  and  the  Equivalent  Sinusoid  of  Exciting  Current. 


sinusoidal  curve  of  magnetic  flux  which  is  setup  in  a transformer 
core,  and  is  the  sinusoidal  curve  of  exciting  current,  lead- 
ing the  curve  by  an  iron  loss  angle  of  18°.  From  these 
curves  the  cyclic  curve  is  constructed  as  heretofore  explained,* 


* Art.  114. 


MUTUAL  INDUCTION,  TRANSFORMERS 


477 


with  its  ordinates  and  abscissas  equal  respectively  to  magnetic 
flux  and  exciting  current.  The  shape  of  the  cyclic  curve 
is  not  that  of  a true  cyclic  curve  of  iron  loss,  as  may  be  seen  by 
the  fact  that  the  maximum  value  of  the  exciting  current  ST 
does  not  occur  at  the  point  of  maximum  flux,  a condition  which 
could  only  occur  in  fact  if  the  eddy  current  part  of  the  iron  loss 
was  relatively  greater  than  is  found  in  commercial  transformers. 
But  if  curve  is  the  equivalent  sinusoid  of  the  actual  irreg- 
ular exciting  current,  the  area  of  RR  must  be  equal  to  the  area 
inclosed  by  the  true  cyclic  curve  of  iron  loss.  It  is  thus  seen 
that  the  substitution  of  an  equivalent  sinusoid  for  the  actual 
curve  of  exciting  current  can  usually  be  made  without  involv- 
ing serious  error. 

126.  The  Effects  of  Variable  Reluctance,  of  Hysteresis,  and  of 
Eddy  Currents  on  the  Form  of  the  Primary  Current  Wave.  — In 

the  preceding  discussions  it  has 
been  assumed  that  the  reluc- 
tance of  the  magnetic  circuit  of 
a transformer  can  be  taken  at 
an  average  constant  value  which 
is  practically  equal  to  the  value 
when  the  current  is  at  its  maxi- 
mum point.  The  low  maximum 
value  of  the  magnetic  density 
which  is  used  in  commercial 
transformers  as  ordinarily  con- 
structed makes  this  assumption 
allowable,  though  it  is  by  no 
means  exact;  and  if  the  magnetic 
density  is  pushed  above  the 
bend  in  the  curve  of  magnetiza- 
tion, the  influence  of  the  low- 
ered permeability  of  the  iron 
becomes  marked.  The  curve 
OM  in  Fig.  272  may  be  taken  to 
represent  the  curve  of  magneti- 
zation of  iron  in  a transformer 
core,  plotted  with  volts  induced 

. . . Fig.  272.  — Diagram  showing  the  Effect 

in  the  primary  Windings  as  or-  0j  Variable  Permeability,  with  Iron 
dinates  and  exciting  ampere-  Loss  Negligible. 


478 


ALTERNATING  CURRENTS 


turns  as  abscissas,  supposing  the  effect  of  hysteresis,  eddy  cur- 
rents, and  magnetic  leakage  to  he  negligible,  and  line  OP  may 
be  taken  to  represent  a corresponding  hypothetical  curve  of  mag- 
netization, assuming  the  effect  of  saturation  to  be  negligible,  i.e. 
the  reluctance  to  be  constant.  Vectors  OB  and  OC  are  the  pri- 
mary and  secondary  ampere-turns  respectively  for  a given  load. 
Then  when  the  magnetizing  ampere-turns  equal  nxI^  represented 
by  OA , the  induced  voltages  in  the  primary  and  secondary  coils 
are  reduced  by  the  effect  of  saturation  from  OF  and  OJ which 
would  be  reached  with  a constant  reluctance,  to  OF'  and  OJ' . 
And,  conversely,  if  the  induced  voltages  are  to  be  OF  and 
OJ',  OA  ampere-turns  are  required  to  bring  the  magnetism  up 
to  the  requisite  value  when  the  curve  of  magnetization  is  031 , 
while  OA!  ampere-turns  would  be  sufficient  if  the  magnetic 
circuits  were  not  influenced  by  saturation. 

The  construction  shows  that  the  saturation  of  the  iron  makes 
necessary  more  turns  of  wire  on  the  primary  and  secondary 
coils  in  order  that  a given  output  may  be  obtained.  Since,  in 
this  case,  the  permeability  varies  through  each  period  with  the 
magnetizing  ampere-turns,  there  is  a periodic  variation  of  Q , and 
the  primary  current  wave  is  distorted  from  the  form  of  the 
primary  impressed  voltage  wave.  This  is  shown  clearly  in 
Fig.  261  of  Art.  114  and  is  there  described. 

When  the  transformer  has  its  secondary  circuit  open,  the 
drop  of  voltage  due  to  the  exciting  current  flowing  through 
the  primary  winding  is  usually  negligible,  so  that  the  pri- 
mary impressed  voltage  at  each  instant  is  proportional  to  the 
tangent  of  the  curve  of  magnetism ; and  is  90°  in  advance  of 
the  magnetization  when  the  latter  is  sinusoidal.  The  tangent 
relations  between  the  curves  of  flux  and  voltage  * must  exist 
as  long  as  IXBX  is  negligible.  The  effect  on  the  output  of 
impressing  excessive  voltages  upon  the  primary  winding  of  a 
transformer  is  indicated  clearly  in  Fig.  272.  Thus,  there  is 
but  slight  rise  of  magnetic  flux  in  the  curve  031  bevond  the 
point  M (near  saturation)  for  increases  of  exciting  current.  If, 
therefore,  OF  is  much  increased,  the  exciting  current  must  in- 
crease excessively.  It  is  thus  seen  that  if  transformers,  or  other 
apparatus  depending  upon  counter-induced  voltage,  are  sub- 
jected to  voltages  that  demand  rise  of  core  flux  beyond  the  point 


* Art.  115. 


MUTUAL  INDUCTION,  TRANSFORMERS 


479 


of  saturation,  the  exciting  currents  demanded  may  reach  danger- 
ous or  destructive  proportions.  Another  important  reason  for 
having  transformers  designed  for  low  flux  densities  in  the  core 
is  because  an  excessive  exciting  current  such  as  spoken  of  above 
will  cause  a material  resistance  drop  in  the  primary,  while  high 
core  reluctance  will  cause  relatively  high  leakage  flux,  thus 
tending  to  interfere  seriously  with  the  regulation. 

As  a very  simple  example  showing  that  the  primary  induced 
voltage  is  always  equal  and  opposite  to  the  impressed  voltage 
when  I1Rl  is  negligible,  and  that  the  exciting  current  varies  in 
such  a way  as  to  furnish  this  induced  voltage  in  a transformer 
with  the  secondary  circuit  open,  we  may  consider  an  inductance 
coil  of  negligible  electrical  resistance  with  a sinusoidal  voltage 
Ex  of,  say,  100  volts  impressed  upon  it.  Suppose  the  frequency 
is  60  periods  per  second,  and  the  self-inductance  Lx  is  .01  of  a 
henry,  then  27rfLx  = 3.71  ohms  is  the  impedance.  The  current 
E 

Ix  flowing  is- — tL_  — 26.5  amperes  with  a lag  angle  of  90°. 

The  induced  voltage  Elm  is  2 irfLxTx  = 100  volts,  which  is  equal 
and  opposite  to  the  impressed  voltage.  Suppose  the  self-in- 
ductance Lx  changes  to  Lx  = .005;  then  the  current  becomes 
i7=53  amperes;  but  the  induced  voltage  EXm  is  2m -fLx'Ix  = 
100  volts,  which  is  the  same  value  as  before.  It  is  seen,  in  this 
case  of  negligible  resistance,  that  whatever  value  the  self-in- 
ductance may  have,  the  exciting  current  takes  such  a magni- 
tude that  the  counter-voltage  of  self-induction  is  numerically 
equal  to  the  impressed  voltage.  Since  the  self-inductance  of 
the  electric  circuit  is  inversely  proportional  to  the  reluctance 
of  the  magnetic  circuit,  when  the  reluctance  changes,  the  excit- 
ing current  changes,  so  that,  as  before,  the  counter-voltage 
equals  the  impressed  voltage.  Figure  262,  described  in  Art. 
114,  shows  that  when  core  losses  and  copper  losses  are  present, 
the  exciting  current  is  advanced  in  phase  so  that  it  is  less 
than  90°  behind  the  impressed  voltage,  and  that  the  summation 
of  the  products  of  the  instantaneous  values  of  impressed  vol- 
tage and  current  is  no  longer  equal  to  zero  for  the  inductance 
coil,  but  becomes  equal  to  the  sum  of  the  core  losses  and  the 
I2R  losses. 

The  effect  of  hysteresis  in  the  core  of  a transformer  is  to  dis- 
tort the  form  of  the  primary  current  wave  to  a still  more  marked 


480 


ALTERNATING  CURRENTS 


degree  than  would  occur  from  the  effects  of  magnetic  saturation 
without  hysteresis,  and  the  higher  the  maximum  magnetic 
density  is  carried,  the  greater  the  distortion  becomes.  The 
ordinates  of  the  primary  current  wave  are  at  each  instant  pro- 
portional to  the  difference  between  the  corresponding  ordinates 
of  the  wave  of  primary  impressed  voltage  and  the  wave  of  in- 
duced counter-voltage.  The  latter  is,  in  an  ordinary  transformer, 
practically  similar  to  the  form  of  the  secondary  voltage  wave. 
With  the  primary  impressed  voltage  sinusoidal  and  the  reluc- 
tance of  the  magnetic  circuit  uniform,  the  primary  current 
wave  would  be  sinusoidal.  With  a variable  reluctance,  hut  no 
hysteresis,  the  current  wave  becomes  peaked,  but  remains  sym- 
metrical; but  when  hysteresis  is  taken  into  account,  the  sym- 
metrical form  is  lost.  This  is  illustrated  in  Figs.  261  and  262.* 

The  effects  which  eddy  currents  in  the  core  have  upon  the 
exciting  current  are  shown  in  Fig.  257,  which  is  described  in 
Art.  112,  and  in  Fig.  263,  described  in  Art.  114.  The  instan- 
taneous value  of  the  portion  of  the  exciting  current  which  is 
required  to  make  up  the  losses  due  to  eddy  currents  at  any 
instant  is  equal  to  the  corresponding  instantaneous  value  of 
the  eddy  current  loss  divided  by  the  instantaneous  value  of  the 
primary  voltage.  The  total  core  loss  maybe  shown  by  plotting 
a cycle  of  the  core  flux  having  abscissas  equal  to  the  arithmetical 
sum  of  the  corresponding  abscissas  of  the  hysteresis  and  eddy 
cycles  (Fig.  263).  The  total  transformer  exciting  current 
may  be  plotted  from  this  as  shown  by  I'  in  Fig.  263. 

The  eddy  current  loss  is  similar  to  that  which  would  be  pro- 
duced by  a closed  secondary  circuit  of  one  turn  of  appropriate 
electrical  resistance,  and  is  unlike  hysteresis  loss,  which  is  de- 
termined by  the  magnetic  quality  of  the  magnetic  core. 
However,  eddy  current  loss  causes  an  increase  in  the  angle  of 
advance  of  the  exciting  current,  as  also  does  current  in  the 
secondary  coil.f  Moreovei’,  the  loss  caused  by  the  exciting 
current  in  the  resistance  of  the  primary  winding  — usually  very 
small — causes  a still  further  advance  of  the  exciting  current.  J 
The  phase  of  the  exciting  current  is  thus  caused  to  be  less  than 
90°  behind  the  impressed  voltage  on  account  of  (a)  the  hyster- 
esis loss,  (5)  the  eddy  current  loss,  and  (e)  the  resistance  loss 
in  the  primary  coil  caused  by  the  flow  of  exciting  current. 

* Art.  114.  t Art.  126.  1 Art.  123  a. 


MUTUAL  INDUCTION,  TRANSFORMERS 


481 


The  angle  by  which  the  exciting  current  is  advanced  from  lagging 
quadrature  with  respect  to  the  voltage  is  called  in  this  volume  the 
Excitation  Angle.  Some  inaccurately  use  the  term  hysteretic 
angle  of  advance  to  express  this  relation,  instead  of  confining  the 
use  of  the  latter  term  to  the  advance  caused  by  hysteresis  alone. 

If  the  eddy  current  cycle  has  a large  area,  its  effect  may 
cause  an  advancement  of  the  instantaneous  maximum  point  in 
the  exciting  current.  This  would  be  the  case,  referring  to  Fig. 
263,  if  the  current  curve  If  had  a maximum  greater  than  the 
magnetizing  current  curve  1^. 

127.  Forms  of  the  Primary  Current  Waves  as  affected  by 
Current  in  the  Secondary  Winding.  — When  the  secondary  cir- 
cuit is  closed,  the  form  of  the  primary  current  is  changed  by 
the  effect  of  the  secondary  current.  In  Fig.  273,  curve  ex  rep- 
resents the  primary  impressed  voltage;  represents  the 

exciting  current  times  the  primary  turns.  Now,  if  by  closing 
the  secondary  cir- 
cuit through  re- 
sistance free  of  re- 
actance, a current 
1, g is  caused  to  flow 
and  the  instanta- 
neous values  of  its 
ampere-turns  are 
represented  by  the 
curve  «2«2  — the 
effects  of  mag- 
netic leakage  are 
here  supposed  to 
be  negligible  — 
the  effect  of  the 
current  flowing  in 
the  secondary  cir- 
cuit is  to  cause  a 
corresponding  increase  in  the  current  flowing  in  the  primary 
circuit.*  The  ampere-turns  of  this  increase  of  the  primary  cur- 
rent may  be  represented  by  curve  nxi^ . The  total  primary 
wave  of  ampere-turns  is  represented  by  curve  n1iv  and  is  the 
sum  of  (the  exciting  ampere-turns)  and  npq'.  The  pri- 

* Art.  118. 

2 1 


Fig.  273.  — Curves  showing  Effect  on  Primary  Current  of  a 
Transformer  caused  by  Current  in  a Non-reactive  Second- 
ary Circuit. 


482 


ALTERNATING  CURRENTS 


mary  current  and  its  components  may  be  directly  shown  from 

this  curve  by  a 
single  change  of 
scale.  It  is  thus 
shown  by  the  fig- 
ure that  the  sec- 
ondary current, 
when  in  phase  with 
the  secondary 
voltage,  tends  to 
reduce  the  distor- 
tion and  lag  of  the 
primary  current. 

If  the  secondary 
circuit  is  induct- 
ive, the  effect  is 
altered  so  that  the 
lag  of  the  primary 
current  is  larger, 
as  shown  in  Fig. 
274.  In  this  case  n2i2  lags  behind  e2  ; and  nxix  also  lags  behind 
ev  The  sum  of  nxix ' and  n^i  , which  gives  nxiv  is  therefore  in 
a lagging  position 
with  respect  to  ev 
the  angle  of  lag  be- 
ing largely  deter- 
mined by  the  angle 
of  lag  of  i2  with  re- 
spect to  e2  where 
is  small  in  compari- 
son with  n2i2. 

If  the  secondary 
current  leads  the 
secondary  induced 
voltage,  the  curve 
corresponding  to  n2i2 
leads  e2,  and  n1i1 
(which  is  the  sum  of  Fig.  275. 


Fig.  274.  — Curves  showing  Effect  on  Primary  Current  of 
a Transformer  caused  by  Current  in  an  Inductive  Second- 
ary Circuit. 


and  nJ  ~)  also 


Experimentally  Determined  Curves  of  Cur- 
rent in  a Transformer  with  Open  Secondary  Circuit. 


leads  ex  when  is  small  in  comparison  with  n2i2. 


MUTUAL  INDUCTION,  TRANSFORMERS 


483 


Figures  275  and 
27 6 show  transformer 
curves  experimental- 
ly observed  by  Pro- 
fessor Ryan  in  1889,* 
using  sinusoidal  im- 
pressed voltage. 
These  show  a strik- 
ing resemblance  to 
the  hypothetical 
curves  built  up  from 
the  loss  cycles.  The 
unmarked  half  sinus- 
oid in  Fig.  275  is  a 
half  cycle  of  the  mag- 
netic wave,  which  is 


Fig.  276.  — Experimentally  Determined  Curves  of  Cur- 
rent in  a Transformer  with  slightly  Inductive  Full 
Load  in  Secondary  Circuit. 

in  lagging  quadrature  with  the  primary  impressed  voltage. 

128.  Core  Magnetization.  — The  maximum  magnetic  density 
during  a period,  in  the  iron  core  of  a transformer,  is  dependent 
upon  the  maximum  value  of  the  magnetizing  component  of  the 
exciting  current  in  the  primary  circuit,  and  is  equal  to 


T = 


4 

10  P 


= 1.257 


P 


where  L is  the  maximum  value  of  the  magnetizing  current 
and  P is  the  reluctance  of  the  magnetic  circuit,  in  gilberts,  at 
the  time  that  the  current  is  maximum. 

Assume,  for  a moment,  that  the  transformer  has  a constant 
reluctance,  no  magnetic  leakage,  and  neither  iron  nor  copper 
loss,  which  is,  of  course,  an  ideal  case.  Then, 


T = 


4V27m1J1_  1 777n1I1 

10  P 1 P ’ 


when  the  secondary  circuit  is  open.  Closing  the  secondary 
circuit  so  that  a current  may  flow  in  it  under  the  impulse  of 
the  induced  secondary  voltage  materially  changes  the  condi- 
tions. We  will  neglect  hysteresis  and  eddy  current  losses  in 
the  iron  core  and  resistance  losses  in  the  windings,  and  first 
assume  that  the  secondary  circuit  is  without  reactance  in  which 

* Trans.  Amer.  Inst.  E ■ E.,  Vol.  7,  p.  1. 


484 


ALTERNATING  CURRENTS 


case  the  secondary  current  I2  will  be  in  unison  with  the  induced 
or  secondary  voltage  E2.  This  current  has  its  own  magnetiz- 
ing effect  on  the  magnetic  circuit.  If  i1  and  i2  are  the  primary 
and  secondary  currents  at  any  instant,  the  total  magneto- 
motive force  in  the  circuit  at  that  instant  is 

4 nT(nxi,  + n2i2) 

10  * 


where  <£>,-  is  the  instantaneous  value  of  the  magnetic  flux,  and  P 
is  the  assumed  constant  value  of  the  reluctance  of  the  magnetic 
circuit.  From  this  is  found 


/10  P4>A 
h ~ V 4 t rnj  ) 


(2) 


If  a sinusoidal  voltage  is  impressed  upon  the  primary  wind- 
ing, which  under  the  assumed  conditions  results  in  sinusoidal 
waves  of  magnetism  and  current,  the  instantaneous  primary 
current  is 

. 10  P<t>  . n2V 2 7, 

i,  — sin  a -cos  a ; 

1 4 irn^  nx 


but 


10 

4 7 rrq 


= V2  7fl, 


whence  ix  = V2 (7^  sin  a — n 2 72  cos  a),  (4) 

ni 

where  a is  measured  from  the  zero  instant  of  the  cycle  of  flux, 
which  in  this  case  is  in  unison  with  IIL  since  losses  are  assumed 
to  be  absent.  Whence 


n1i1  = sin  a — n2I2  cos  «), 

and  ( nfl sin2  a 

— 2 nxn2I^I2  sin  « cos  a 
+ n2I2  cos2  a) da,  (4) 

where  7,  is  the  effective  value  of  the  primary  current  when  the 
secondary  current  is  equal  to  72 ; and  as  before,  is  the  watt- 
less primary  current  when  the  secondary  circuit  is  open.  Per- 
forming the  integration  gives 

n2i2=n*Il?  + n*I*. 


(5) 


MUTUAL  INDUCTION,  TRANSFORMERS 


485 


Remembering  that  I and  /2  have,  in  this  case,  90°  difference  of 
phase,  the  three  terms  of  this  formula  may  be  represented  by 
the  three  sides  of  a right-angled  triangle,  as  in  the  triangle 
formed  by  the  points  OAB  in  Fig.  272,  in  which  excitation 
losses  are  assumed  absent.  The  current  Ix  amperes,  which 
flows  in  the  primary  circuit  when  the  secondary  circuit  is 
loaded,  is  in  advance  of  the  current  1^  by  an  angle  the  tan- 


1 " is  here  a wattless  magnetizing  component  of  the  primary 
current,  losses  being  absent  by  assumption,  it  is  directly  de- 
pendent on  the  reluctance  of  the  magnetic  circuit  and  the  coun- 
ter-voltage in  the  primary  coil.  As  the  reluctance  is  usually 
small,  I is  small. 

When  the  transformer  has  an  iron  core,  the  exciting  current 
is  no  longer  sinusoidal  or  in  exact  leading  quadrature  with  the 
secondary  induced  voltage,  but  is  distorted  and  is  advanced  by 
the  amount  of  the  excitation  angle.*  Call  this  angle  77,  and  con- 
sider it  the  advance  of  the  equivalent  sinusoid  substituted  for 
the  irregular  curve  of  exciting  current.  This  substitution  can 
be  made  with  sufficient  accuracy  for  this  discussion.  The 
magnetism  remains,  as  before,  90°  ahead  of  the  induced  vol- 
tage although  reluctance  varies,  so  that  the  following  relations 
obtain : 


This  is  the  formula  of  a triangle  where  90°  — 77  is  the  angle 
included  between  the  sides  representing  and  — n2Iv  such  as 


10P<3>  . 


iUjrcp  . /or-/  , \ 

sin  « = V 2 1 sin  (oc  + 77), 

4 7 r/q 


(6) 


being  the  primary  exciting  current. 
Substituting,  as  before,  in  (3)  gives 


IT  Jo 

— 2 n^I  I2  sin  (a  + 77)  cos  a 
-f-  n22I22  cos2  a)  da, 


or, 


«i2A2  = ni%2  + n‘ili  - 2 sin  V 

= zq2//2  + «22Z22 - 2 cos  (90°  - 77) . (7) 


* Art.  125. 


486  ALTERNATING  CURRENTS 

OA  and  AB  of  the  triangle  OAB  of  Fig.  266,  in  which  the 
effect  of  excitation  losses  is  shown. 

129.  The  Circle  Diagram  for  Non-reactive  Secondary  Circuit 
and  Active  Voltage  Locus.  — In  a constant  voltage  transformer 
the  load  is  zero  when  the  resistance  of  the  secondary  circuit  is 
infinite,  and  the  load  increases  as  the  secondar}7  resistance  de- 
creases through  the  range  of  normal  currents  for  which  the 
transformer  is  designed.  Then,  if  leakage  reactance  is  con- 
sidered constant  for  all  loads,  as  may  be  done  without  appre- 
ciable error  when  the  frequency  of  the  exciting  current  is  con- 
stant, its  relative  effect  in  causing  voltage  drop  will  increase  with 
increase  of  load.  These  relations  may  be  neatly  shown  by  a con- 
struction similar  to  that  discussed  before.*  Thus,  in  Fig.  277, 

OC represents  the  impressed 
voltage  of  a transformer  in 
which,  for  convenience,  a 
unity  ratio  of  transforma- 
tion is  assumed.  The  right- 
angled  triangle  OG-C  is 
then  constructed  on  OC. 
The  side  GO  is  made  equal 
to  the  drop  of  impressed 
voltage  due  to  the  reactive 
voltage  CF  of  the  primary 
coil  and  FG  of  the  second- 
ary coil  or  — G C = El  = 
ElLl  + The  side  OG 

is  made  equal  to  the  sum  of 
J G , the  voltage  expended 
in  the  resistance  of  the 
primary  coil  Rx  on  account 
of  the  flow  of  the  current 
//,  which  is  equal  and 
opposite  to  the  secondary 

current  _£>,  the  voltage  KJ \ 
AD  2 ° ’ 

Fig.  277.  - Transformer  Diagram  for  a Trans-  which  is  e(lual  and  Opposite 
former  containing  Leakage  Reactance  and  to  the  voltage  expended  in 
Non-reactive  Secondary  External  Circuit.  the  secondary  coil  resistance 

Rv  and  the  voltage  OK , which  is  equal  and  opposite  to  the 


/ 


* Art.  85. 


MUTUAL  INDUCTION,  TRANSFORMERS 


487 


voltage  expended  in  the  load  resistance  R.  In  obtaining  these 
voltage  drops  the  exciting  component  of  the  primary  current 
is  neglected  for  simplicity,  as  its  effect  is  ordinarily  small. 
Therefore 

OG  = EyR<  = — /2  (Ry  + R2  + R)  = ly  ( /('j  + R2  + R), 

the  ratio  of  transformation  being  taken  as  unity.  The  sides 
OG-  and  GC  thus  form  the  right-angled  triangle  of  active  and 
reactive  voltages  in  the  transformer. 

Since  GC  increases  with  the  current  and  angle  OGC  must  re- 
main aright  angle  for  any  current,*  the  locus  of  the  point  G , 
which  moves  as  the  current  changes,  is  the  arc  of  a circle  with 
OC  as  its  diameter.  In  a well-made  commercial  transformer 
with  its  primary  and  secondary  coils  divided  into  parts  and  sand- 
wiched together  to  avoid  leakage,  the  line  GC  is  relatively  very 
short  even  at  full  load,  so  that  the  point  G travels  only  a short 
distance  from  C as  the  secondary  current  is  increased  from  zero 
to  the  full  load  current.  The  point  G would  be  at  the  point 
C for  all  loads  if  the  load  and  coil  reactances  were  zero. 

The  secondary  terminal  voltage  KO  = RIV  the  voltage  in  the 
resistance  JK=  E2R^  or  the  reactance  voltage  FG  = E2U  of  the 
secondary  coil  may  readily  be  obtained  for  any  ratio  of  trans- 
formation by  merely  using  a scale  equal  to  - times  that  used  in 

s 

the  original  construction  and  reversing  the  secondary  circuit 
voltage  or  current  arrow  shown  in  the  figure.  The  total  in- 
duced voltage  due  to  the  mutual  magnetism  of  the  primary 
and  secondary  windings  is  HO  = E2m  = E' lm  = —Elm,  the  ratio 
of  transformation  being  unity.  The  voltage  drop  in  the  primary 
coil  resistance,  due  to  the  current  component  opposite  to  the  sec- 
ondary current,  is  JG  =HF=  E1R , and  the  voltage  drop  by  rea- 
son of  the  primary  coil  reactance  is  EC  = ElLi. 

The  component  of  primary  current  used  for  excitation  is  indi- 
cated by  the  line  AO , and  is  composed  of  an  active  component 
DO,  in  phase  with  OC  and  a quadrature  component  AD  in  phase 
with  the  core  magnetism.  The  component  D 0 is  of  such  value 
that  when  multiplied  by  OC ',  the  product  equals  the  excitation 
losses.  The  exciting  current  is  assumed  for  the  purpose  of  the 
diagram  to  be  constant  in  scalar  value  and  phase  position,  which 


* Art.  85. 


488 


ALTERNATING  CURRENTS 


is  sufficiently  accurate  for  most  practical  cases.  Its  assumed 
constant  value  is  usually  taken  as  the  amperes  flowing  in  the 
primary  circuit  when  the  secondary  circuit  is  open,  i.e.  the 
Open  circuit  current.  On  the  other  hand,  the  current  I2  of  the 
secondary  coil  and  its  opposing  element  I\  of  the  primary 
vary  directly  with  the  load  or  inversely  with  the  impedance. 
For  the  present  we  will  continue  to  consider  the  load  non-reac- 
tive and  the  leakage  reactances  constant.  Then  we  have  a case 
equivalent  to  a circuit  having  variable  resistance  and  fixed  in- 
ductive reactance.  In  such  a circuit,  as  was  proved  earlier,* 
the  locus  of  the  current  vector  is  a semicircle,  located  in  the 
first  trigonometrical  quadrant,  with  its  diameter  on  the  X-axis, 
one  end  being  at  the  origin.  The  diameter  of  this  semicircle 
was  shown  to  be  equal  to  the  impressed  voltage  divided  by  the 
reactance.  In  Fig.  277,  OC  may  correctly  be  used  as  the  vol- 
tage absorbed  in  the  primary,  secondary,  and  load  impedances, 
by  the  flow  of  the  secondary  current,  or  its  primary  opposing 
component,  as  heretofore  explained.  Then  a semicircle  with 
its  center  on  OV  extended,  as  shown  at  OBT \ with  a diameter 
equal  to  OC  divided  by  the  combined  leakage  reactances 


m 


, that  is,  equal  to 


K 


„ , is  the  locus  of  the  end  of 

Xiu+Xu. 

OB  ( = J71)  or  of  the  secondary  current  vector  reversed.  In 
this  case,  X,  the  reactance  of  the  external  secondary  load  is 
assumed  to  be  zero,  that  is,  the  load  is  non-reactive.  When 
the  primary  triangle  of  voltages  is  OGC,  the  secondary  current 
is  B 0,  which  is  in  phase  with  the  active  element  of  the  voltage, 
JO  = X2(/?2  + ^).  The  primary  current  is  then  the  vector  AB  = 
AO  + OB.  The  secondary  current  can  be  read  directly  from 
the  diagram  for  any  ratio  of  transformation  by  using  a scale 
with  s times  as  many  amperes  to  the  inch  as  that  used  in  the 
construction. 

If  the  impressed  voltage,  kilowatt  capacity,  exciting  current 
as  a percentage  of  full  load  current,  core  losses  in  percentage 
of  kilowatt  capacity,  and  the  resistances  and  reactances  of  the 
coils  and  load  of  a transformer  are  known,  the  construction, 
such  as  shown  in  Fig.  277,  can  at  once  be  made.  From 
this  the  primary  and  secondary  currents  are  scaled  off  on 
AB  and  BO  in  amperes.  The  angle  of  lag  of  the  primary 


*Art.  70,  Case  (1). 


MUTUAL  INDUCTION,  TRANSFORMERS 


489 


current  is  the  angle  B WO.  The  angle  of  lag  of  the  sec- 
ondary current  is  the  angle  between  the  terminal  voltage 
KO  = E%  and  the  current  BO  = IV  which  angle  is  zero  under 
the  circumstances  here  assumed.  The  per  cent  regulation  for  the 
given  load  is  one  hundred  times  the  ratio  of  00—  OK  to  OK,  as- 
suming unity  ratio  of  transformation.  The  various  reactance  and 
resistance  drops  are  measured  along  G-0  and  0G-  respectively. 
The  power  received  from  the  mains  is  00  times  the  projec- 
tion of  AB  thereupon,  or  W{  = EXIX  cos  6V  The  power  absorbed 
in  the  load  is  KO  times  BO  or  W„=  E^IV  which,  though  ob- 
tained in  the  diagram  for  a transformer  with  ratio  of  transfor- 
mation (s)  equal  to  unity  is  also  true  without  change  when  s has 
any  other  value,  since  the  primary  equivalent  voltage  is  divided 
by,  and  the  current  multiplied  by,  s to  reduce  to  secondary 
values.  The  total  power  losses  in  the  transformer  are  : exciting 
current  losses  BO  x 00;  the  primary  coil  copper  loss  due  to 
current  OB,  JO  x OB  ; and  the  secondary  copper  loss,  JK x BO. 
The  sum  of  these  losses  subtracted  from  the  power  input  equals 
the  power  output.  The  commercial  efficiency  of  the  trans- 

IT 

former  is  the  total  output  divided  by  the  total  input  or  77  = — ° . 

W i 

Changing  the  position  of  B to  various  points  on  its  locus 
OBT  and  moving  O along  the  semicircle  OMO  so  that  OO  is 
kept  in  the  same  direction  as  OB,  permits  the  determination  of 
the  entire  transformer  performance  for  any  value  of  the  secon- 
dary current  or  fraction  of  the  rated  capacity  of  the  trans- 
former. 

When  the  reactive  voltage  OO  is  very  short,  as  in  the  case  of  a 
well-made  commercial  transformer  supplying,  for  example,  an  in- 
candescent lamp  circuit,  the  line  OKJO  almost  coincides  with 
the  line  00.  In  this  case  the  locus  OBT  has  such  a large  diame- 
ter that  it  is  almost  a vertical  straight  line  from  no  load  to  full 
load  ; then,  if  OA,  for  the  particular  character  of  the  calcula- 
tion in  question,  may  be  considered  of  negligible  length,  KO, 
AB,  and  BO  may  be  considered  without  serious  error  to  vary 
with  the  load  up  and  down  the  line  00,  and  the  calculation  of 
the  transformer  regulation  resolves  itself  into  the  simple  prob- 
lem of  three  resistance  drops  in  series.  The  three  resistances 
causing  these  drops  are  the  fixed  resistances  of  the  primary  and 
secondary  coils  and  the  variable  resistance  of  the  load. 


490 


ALTERNATING  CURRENTS 


In  the  foregoing  discussion  AO  was  considered  constant. 
Though  approximately  true  this  is  not  exactly  the  case,  as  the 
magnetizing  current  must  be  proportional  to  the  total  mutually 
induced  voltage  HO,  and  leading  it  by  an  angle  of  90°  plus  the 
excitation  angle  of  advance.  An  approximately  accurate  cor- 
rection to  calculations  can  be  made,  when  necessary,  by  giving 
A 0,  for  each  load,  such  a length  that  it  will  have  the  same 
ratio  to  OH  that  the  open  circuit  current  has  to  00,  and  also 
turning  it  to  such  a position  that  it  maintains  the  constant 
angular  relation  to  OH.  Inasmuch  as  the  construction  of  the 
length  OH  is  dependent  upon  the  resistance  drop  HF,  the 
length  and  phase  of  HF  must  be  corrected  if  great  accuracy 
is  desired  so  as  to  include  the  drop  of  voltage  due  to  exci- 
tation losses.  The  diagram  presupposes,  also,  that  the  reluctance 
of  the  core  is  constant  for  all  loads,  and  that  the  leakage  paths 
are  free  from  iron  losses.  These  assumptions  for  commercial 
transformers  introduce  no  great  error.  The  further  assumption 
that  the  voltages  and  currents  are  sinusoidal  may,  under  some 
circumstances,  require  correction,  as  when  resonance  is  present 
for  some  of  the  harmonics.  The  effects  of  irregular  waves  of 
voltage  and  current  will  be  treated  later.  For  ordinary  trans- 
former calculations  it  is  usual  to  assume  that  the  voltage  and 
currents  are  sinusoidal. 

130.  Circle  Diagram  where  the  Transformer  Load  contains 
Constant  Reactance  and  Variable  Resistance. — Suppose  now, 
without  changing  the  unity  ratio  of  transformation  or  other  as- 
sumptions, that  the  load  contains  inductive  reactance  X,  then 
the  construction  shown  in  Fig.  277  may  be  conveniently  modi- 
fied as  in  Fig.  278.  Here  the  component  of  the  impressed 
voltage  OC,  used  in  overcoming  the  total  reactive  voltage,  is  QC, 
in  which  OF  and  F(J  are  the  primary  and  secondary  coil  re- 
active voltages  respectively  and  GrQ  the  load  reactive  voltage 
transferred  to  the  primary  circuit.  The  total  active  voltage 
in  phase  with  the  component  of  the  primary  current  which  is 
equal  and  opposite  to  the  current  in  the  secondary  coil  is  OQ. 
This  is  expended  in  the  resistance  of  the  coils  and  load.  The 
secondary  terminal  voltage  is  KO , and  is  composed  of  KK , 
which  is  equal  and  opposite  to  the  reactive  voltage  of  the  load, 
and  K'O,  which  is  the  voltage  drop  in  the  load  resistance. 
The  secondary  angle  of  lag  02  the  angle  X' OK.  The  dia- 


MUTUAL  INDUCTION,  TRANSFORMERS 


491 


gram  is  similar  in 
construction  to  that 
of  Fig.  277,  and  its 
further  description 
and  use  may  be  ob- 
tained by  the  pre- 
ceding discussion 
concerning  the  latter. 

Electrostatic  ca- 
pacity in  the  load 
tends  to  neutralize 
the  self-inductance 
of  the  primary  and 
secondary  coils.* 
Thus,  if  capacity  is 
introduced  into  the 
secondary  circuit  of 
such  value  that  the 
capacity  voltage 


equals  the  inductive  ^78. — Transformer  Diagram  showing  the  Effect  of 
1 _ . _.  Inductive  Reactance  in  the  Secondary  External  Circuit. 

voltage  (xQ  m big. 

278,  the  reactive  voltage  remaining  is  (7(7,  and  the  construction 
becomes  like  Fig.  277.  If  the  capacity  increases,  the  point  (7 
moves  toward  (7,  reaching  it  when  the  capacity  voltage  equals 
the  inductive  voltage  CQ.  Because  of  this  increase  of  capacity, 
the  diameter  of  the  current  locus  OB T changes  from  the  value 


X, 


XUl  + 


to  the  value 


A. 


xlLi+x^+xL-xc 


, where  Xc  is 
E 


the  capacity  reactance,  and  the  diameter  becomes  equal  to  — = 


when  the  capacity  reactance  in  the  denominator  is  equal  to  the 
sum  of  the  inductive  reactances.  For  each  increase  in  capacity 
a locus  corresponding  to  OBT  must  be  drawn  with  a diame- 
ter larger  than  the  last,  until  the  capacity  balances  the 
self-inductance,  when  the  diameter  becomes  infinite  so  that  the 
arc  of  the  circle  coincides  with  00.  When  the  capacity  vol- 
tage exceeds  CQ,  the  system  acts  as  one  having  resistance  and 
capacity  alone,  and  the  current  takes  a leading  angle  determined 
by  the  excess  of  capacity  reactance;  then  the  point  Q has  a 


* Arts.  81  and  82. 


492 


ALTERNATING  CURRENTS 


locus  on  the  left  of  OC , as  shown  in  Fig.  279.  Thus  the  current 
locus  is  a semicircle  to  the  left  of  OC,  such  as  OB^T',  and  has 
a diameter  equal  to  the  impressed  voltage  divided  by  the  net 
excess  of  capacity  reactance.  This  is  in  accord  with  the 
theorem  earlier  proved  * for  a circuit  having  constant  capacity- 
reactance  and  variable  resistance.  Otherwise  the  diagram  is 
essentially  as  in  the  case  of  an  inductively  reactive  circuit,  ex- 
cept that  the  primary  current  is  somewhat  smaller  and  the 
angle  of  lead  somewhat  less  than  would  have  been  the  current 
and  lag  angle  for  an  equal  net  amount  of  inductive  reactance 
instead  of  capacity  reactance.  This  is  because  the  exciting 
component  of  the  current,  which  need  move  its  vector  position 
only  slightly  under  ordinary  conditions  and  is  here  considered 
stationary,  is  in  a different  phase  position  in  the  two  instances 
with  relation  to  the  remaining  component  of  the  primary 
current. 

Figure  279  shows  the  conditions  assumed  above  when  the 
secondary  current  is  of  the  same  value  as  in  Figs.  277  and  278. 
The  lettering  is  similar  in  the  three  figures.  In  Fig.  279  is 
shown  the  impressed  voltage  0(7;  the  active  voltage  locus  for 
inductive  and  capacity  reactances,  which  is  the  circle  OMCN ; 
the  current  locus  OBT  where  the  net  predominance  of  inductive 
voltage  is  CQ,  as  in  Fig.  278  ; the  current  locus  OB^T  where 
the  net  predominance  of  capacity  voltage  is  CQQ  ; and  the 
current  locus  OCT"  when  inductive  and  capacity  voltages 
neutralize  each  other  or  are  both  negligible. 

To  determine  the  lengths  and  phase  positions  of  the  lines  rep- 
resenting the  vectors  of  mutually  induced  primary  voltage  and 
the  secondary  terminal  voltage,  take,  for  example,  the  internal 
and  load  reactive  voltages  of  Fig.  278,  which  are  reproduced  in 
the  line  CQ  of  Fig.  279,  and  suppose  a net  excess  of  capacity 
reactive  voltage  is  represented  by  the  line  CQJ.  To  give  this 
excess,  the  total  capacity  voltage  must  then  he  equal  to  CQ / 
plus  a voltage  equal  and  opposite  to  the  inductive  voltage. 
The  point  QQ  is  fixed  by  the  length  QQ  C and  the  locus  of  the 
active  voltage  ONC ; therefore,  the  total  capacity  voltage  may 
he  laid  out  from  QJ  to  Q',  the  length  of  CQ'  being  equal  to 
CQ  but  its  direction  being  in  quadrature  with  the  current  OB x' 
and  therefore  in  line  with  Q^  C.  The  part  of  CQ'  represented 


Art.  70,  Case  (1). 


MUTUAL  INDUCTION,  TRANSFORMERS 


493 


by  G'Q'  neutralizes  the  effect  on  the  transformer  of  the  load 
inductance.  The  triangle  of  voltages  at  the  secondary  termi- 
nals is  then  OK^ K'  and  the  impressed  voltage  OC  may  be  con- 
sidered to  be  applied  in  the  following  components  : OK^ , which 


Fig.  279.  — Transformer  Locus  Diagram  showing  the  Effects  of  Capacity  or  Induc- 
tance Predominant  or  Equal. 

is  equal  and  opposite  to  the  secondary  active  voltage ; K' 
which  is  equal  and  opposite  to  the  secondary  capacity  voltage 
in  excess  of  the  inductive  voltage  of  the  load ; K'J the  drop 
equal  and  opposite  to  the  drop  in  the  secondary  coil  resistance ; 
J1  H' , the  drop  equal  to  the  secondary  leakage  voltage ; H'F’,  the 


494 


ALTERNATING  CURRENTS 


drop  in  the  primary  coil  resistance ; and  F'O,  the  drop  equal 
and  opposite  to  the  primary  leakage  voltage.  The  total  pri- 
mary mutually  induced  voltage  is  H'  0 and  the  total  secondary 
terminal  voltage  is  K'  0.  The  coil  resistance  drops  may  be 
transferred  from  K J'  Gr'  to  the  active  voltage  line  at 
and  the  diagram  then  becomes  similar  to  those  previously  dis- 
cussed. The  secondary  angle  of  lead  is  the  angle  K'  OK j'  and 
the  primary  angle  of  lead  is  the  angle  between  the  lines  CO 
and  B^A  extended. 

When  inductive  and  capacity  reactances  neutralize,  as  when 
the  current  locus  is  OCT ",  the  line  Q^Q'  swings  to  a horizontal 
position  and  Q^C=CQ'  ; the  coil  impedance  triangle  CGr'K' 
therefore  swings  down  so  that  its  side  CG-'  is  horizontal.  It  is 
thus  seen  from  Fig.  279  that  for  a given  angle  between  the 
current  and  impressed  voltage  the  value  of  H'  0 is  greater  when 
the  capacity  reactance  is  in  excess  compared  with  its  value 
when  inductive  reactance  is  in  excess,  but  for  commercial  con- 
stant voltage  transformers  operating  with  ordinary  power  factors 
and  normal  secondary  currents,  the  change  of  H’  0 , the  mutually 
induced  primary  voltage,  in  either  length  or  position  is  not 
great.  When  neither  inductance  nor  capacity  is  present,  H'  0 
swings  into  the  line  0(7,  and  in  practice  it  usually  remains 
close  thereto. 

The  circular  arcs  0T2,  0T2',  and  0T2"  are  the  circular  loci 
that  would  be  made  by  the  secondary  currents  had  they  been 
drawn  outward  from  0 instead  of  inward  as  shown  in  Fig. 
279.  When  these  arcs  are  used,  a phase  instead  of  a vector  dia- 
gram of  the  currents  results. 

Prob.  1.  Given  a 60-cycle  transformer  of  100  kilowatts  rated 
capacity,  in  which  the  primary  voltage  is  2200  volts,  the  ratio 

of  transformation,  ^l,  is  20,  the  copper  loss  at  full  load  2 per 
n2 

cent  of  the  rated  capacity  and  equally  divided  between  the 
primary  and  secondary  coils,  the  iron  loss  1 per  cent  of  the 
transformer  rated  capacity,  the  exciting  current  (assumed  con- 
stant) 2 amperes,  and  the  drop  of  voltage  at  full  load  due  to 
magnetic  leakage  1 J,  per  cent  of  the  impressed  voltage  when 
current  is  being  delivered  to  a non-reactive  secondary  circuit  : 
Draw  for  this  transformer  a large  transformer  diagram  similar 
to  Fig.  277. 


MUTUAL  INDUCTION,  TRANSFORMERS 


495 


(Note.  — For  convenience  use  secondary  quantities  in  terms  of  primary 
equivalents  in  making  the  diagram.  After  its  completion  read  off  the  correct 
secondary  quantities  by  changing  the  scale.  The  point  G for  full  load  may 
be  found  by  laying  out  GC  so  that  OG  is  1’  per  cent  less  in  length  than  OC. 
Since  OG  is  the  active  component  of  OC  it  is  proportional  to  the  terminal 
voltage  plus  the  2 per  cent  allowed  for  copper  losses,  and  KG  is  therefore 
0° 

per  cent  of  OG,  while  KO  is  the  primary  equivalent  of  the  secondary 


terminal  voltage  E2  at  the  full  load  of  100  kilowatts.  Divide  100,000  (the  load 
in  watts)  by  the  number  giving  the  volts  represented  by  the  scale  length  of 
KO,  and  the  value  of  amperes  in  the  full  load  secondary  current,  — //, 
is  obtained,  expressed  in  primary  equivalents.  Then  the  total  primary 
equivalent  leakage  reactance  XL  is  approximately  the  volts  CG  divided  by 


and  the  diameter  of  the  current  locus  is  . Make  01)  of  a length 

which  represents  in  amperes  that  number  which  wdien  multiplied  by  the 
volts  represented  by  OC  gives  one  per  cent  of  the  transformer’s  rated  capac- 
ity. This  fixes  the  phase  angle  of  OA  under  the  assumption  that  the  mag- 
netizing element  of  the  current  is  constant  in  ATalue  and  at  right  angles  to 


OC.) 


Assuming  the  exciting  current  constant  in  length  and  phase 
position,  find  the  primary  and  secondary  currents,  the  secondary 
terminal  voltage,  the  primary  and  secondary  angles  of  lag  and 
the  regulation  (ratio  of  primary  voltage  minus  secondary  ter- 
minal voltage  in  terms  of  the  primary  to  primary  voltage)  for 
full  load  and  for  25  per  cent,  50  per  cent,  75  per  cent,  and  125 
per  cent  of  full  load. 

Prob.  2.  Assume  the  same  specifications  as  in  Prob.  1 for 
the  transformer  and  conditions  there  specified,  except  that  the 
power  factor  of  the  load  has  become  0.9  on  account  of  the  in- 
troduction of  self-inductance.  Draw  the  diagram  and  compute 
the  currents,  voltages,  angles  of  lag,  and  regulation  as  required 
in  Prob.  1.  Also  give  the  amount  (in  henrys)  of  self-inductance 
introduced  into  the  load. 

Prob.  3.  Assume  the  same  specifications  as  in  Prob.  1 except 
that  the  secondary  circuit  contains  a net  capacity  reactance,  so 
that  the  power  factor  of  the  load  is  .9,  with  current  leading. 
Draw  the  diagrams  and  compute  the  currents,  voltages,  angles 
of  lag,  and  regulation  as  required  in  Prob.  1.  Find  the  amount 
(in  microfarads)  of  capacity  in  the  secondary  circuit. 

131.  Short  Circuits.  — As  explained  in  the  preceding  articles, 
the  point  6r  in  a transformer  diagram,  such  as  Fig.  277,  does 


496 


ALTERNATING  CURRENTS 


not,  for  the  ordinary  working  power  factors  and  loads  used  in 
commercial,  constant  voltage  transformers,  move  far  from  the 
point  C,  within  the  practicable  ranges  of  continuous  load.  If, 
however,  for  any  reason,  the  terminals  of  the  secondary  winding 
are  connected  together — short-circuited  — through  a negligible 
resistance,  the  secondary  terminal  voltage  KO  becomes  zero 
and  Gr  swings  around  the  semicircle  until  iTlies  on  0.  This 
means  that  the  primary  and  secondary  currents  may  become 
enormous  compared  with  the  maximum  currents  for  which  the 
transformer  was  designed  and  will  be  approximately  equal  to 
Hj 

-l,  where  Ex  is  the  impressed  voltage,  and  Zx  is  the  combined 

A 

impedance  of  the  primary  and  secondary  coils  — the  latter  re- 
duced to  primary  equivalents. 

The  conditions  are  represented  in  Fig.  280,  where  the  current 
locus  is  OBB' T,  the  voltage  locus  OMCN , the  rated  full  load 


Fig.  280.  — Transformer  Diagram  showing  Conditions  when  the  Secondary  Termi- 
nals are  Short-circuited. 


secondary  current  BO , the  primary  and  secondary  impedance 
voltage  triangles  HFC  and  KJH  respectively,  the  diagram 
being  lettered  to  correspond  with  Fig.  277.  All  secondary 
quantities  are  shown,  as  heretofore,  as  primary  equivalents  : 
and,  as  before,  the  coil  impedances  are  exaggerated  for  the  sake 
of  clearness  in  the  diagram. 

When  the  secondary  terminals  are  short-circuited,  Gr  moves 
to  Gr1  and  the  entire  impressed  voltage  OC  is  expended  in 


MUTUAL  INDUCTION,  TRANSFORMERS 


497 


driving  current  through  the  coil  impedances  as  shown  by  the 
triangles  H'F'  C and  K'J'H'.  The  secondary  current  is  B'  0, 
which  is  several  times  larger  than  BO.  The  mutually  induced 
primary  voltage  here  becomes  H'  0 , which  is  out  of  phase  with 
HO  and  is  much  shorter.  Since  H'  0 must  be  proportional  to 
the  magnetic  flux  which  is  mutual  to  the  primary  and  secondary 
windings,  this  means  that  the  mutual  flux  in  the  core  decreases 
with  the  increase  of  magnetic  leakage  and  resistance  drop  caused 
by  the  extraordinarily  large  currents  and  the  accompanying  large 
magneto-motive  forces  impressed  on  the  leakage  paths.  Exciting 
current  AO  can  no  longer  be  considered  constant  but  must  be 
changed  by  reducing  its  vertical  (active)  component  in  a ratio 

approximately  equal  to  6 •>  and,  supposing  that  the  core 

reluctance  remains  approximately  constant,  reducing  its  hori- 
zontal (reactive)  component  in  the  ratio  of  • It  is  then 


A'O  and  the  primary  current  is  A'B' . 

For  the  sake  of  making  a clear  diagram  of  small  dimensions, 
the  leakage  and  copper  drops  have  been  greatly  exaggerated  in 
Figs.  277,  278,  279,  and  280,  as  compared  with  the  values  they 
would  have  in  large,  well-designed,  commercial  transformers  in- 
tended for  constant  voltage  working.  In  such  transformers 
the  radius  of  the  semicircle  OBB' T would  be  many  times 
larger  than  is  shown  in  Figs.  277,  278,  279,  and  280,  and  the 
point  B would  move  farther  along  the  semicircle  towards  T in 
case  of  short  circuit,  so  that  OB'  may  become  enormously  larger 
in  proportion  to  the  full  load  current  than  is  shown  in  Fig.  280. 

The  coils  of  a transformer  are  usually  wound  close  together, 
as  seen  in  Fig.  46,  or  are  sandwiched  together  in  sections,  to 
prevent  undue  magnetic  leakage,  and  are  bound  and  supported 
only  by  insulating  materials.  When  a short  circuit  occurs,  the 
forces  tending  to  repel  the  primary  and  secondary  coils  one 
from  the  other,  by  reason  of  th'e  large  currents  and  excessive 
leakage  fluxes  (proportional  to  F'C  and  Gr'  F'  in  Fig.  280), 
may  become  very  great.  The  primary  and  secondary  currents 
being  substantially  in  opposition,  the  forces  set  up  between  them, 
by  virtue  of  their  magnetic  leakage  fluxes,  are  of  repulsion. 
Also,  the  leakage  fluxes  being  proportional  to  the  currents  in 
primary  and  secondary  coils,  this  repulsion  is  proportional  to 
2k 


498 


ALTERNATING  CURRENTS 


the  product  of  the  currents,  and  as  the  secondary  and  primary 
currents  of  a transformer  approximately  vary  together,  the 
mechanical  force  exerted  between  the  coils  is  closely  propor- 
tional to  the  square  of  the  current  flowing  in  either  winding. 

The  mechanical  strains  thus  set  up  in  a small  transformer 
which  becomes  short-circuited  may  not  exceed  the  strength  of 
the  insulating  supports  of  the  coils.  In  the  case  of  large  trans- 
formers the  short-circuit  currents  become  larger  in  proportion 
to  the  rated  full  load  currents,  because  of  the  proportionately 
lower  resistances  of  the  windings  of  the  larger  machines,  and  the 
fluxes  are  larger,  hence,  the  electro-magnetic  stresses  may  be- 
come very  great  if  short  circuits  occur.  For  transformers,  of 
similar  physical  form  but  different  sizes,  the  stresses  referred 
to  may  increase  in  proportion  to  the  squares  of  the  rated 
loads;  so  that  of  two  transformers,  one  of  which  has  a capacity 
of  10  kilowatts  and  the  other  of  1000  kilowatts,  the  repulsion 
strain  on  conductors  of  the  second  may  be  in  the  neighborhood 
of  10,000  times  as  great  as  in  the  first.  The  result  is  that  the 
windings  of  very  large  transformers  are  sometimes  torn  to 
pieces,  as  by  an  explosion,  when  the  secondary  circuits  are  acci- 
dentally short-circuited. 

The  burning  out  of  the  coils,  due  to  the  excessive  I2R  losses, 
when  a short  circuit  is  maintained,  will  occur  as  in  any  other 
electrical  machine  or  circuit,  but  injury  from  repulsion  may  oc- 
cur before  the  circuit  is  broken  by  a burn-out  or  by  the  action 
of  an  automatic  circuit  breaker.  The  computation  of  the 
electro-magnetic  stresses  has  been  attempted  and  may  be  ac- 
complished to  some  degree  of  approximation.* 

132.  Locus  of  Current  in  a Transformer  when  the  Load  Re- 
actance is  varied  and  the  Resistance  is  kept  Constant.  Relation 
of  Power  Factor  to  Capacity.  — In  the  discussions  just  preceding, 
the  load  and  coil  reactances  of  a constant  voltage  transformer 
were  considered  to  be  constant  while  the  resistance  varied. 
It  is  also  possible  to  have  a load  in  which  the  resistance  is  ap- 
proximately constant  while  the  power  factor  varies  because  of 
changing  inductive  or  capacity  reactance,  as  when  the  load 
comprises  a synchronous  motor  operating  at  constant  torque 
but  with  various  field  excitations,  f Consider  a case  of  the 
latter  kind  and  assume  the  transformer  to  have  negligible  mag- 


* Trans.  Amer.  Inst.  E.  E.,  Vol.  30. 


t Art.  160. 


MUTUAL  INDUCTION,  TRANSFORMERS 


499 


netic  leakage.  Then  let  00  in  Fig.  281  be  the  impressed  vol- 
tage. The  voltage  locus  is  OMON \ as  before.  Consider  first 
that  the  load  and  coil  reactances  are  zero.  Then  the  voltages 
of  the  primary  and  secondary  coils  induced  by  the  mutual  flux 
lie  on  the  line  0(7,  if  the  slight  deviation  therefrom  due  to 


Resistance  is  maintained  Constant. 

the  small  element  of  impressed  voltage  in  phase  with  the  ex- 
citing current  is  neglected.  The  line  00  represents  the  im- 
pressed voltage  on  the  primary  coil ; HO  is  the  primary  voltage 
drop  in  the  primary  coil  resistance  ; KH  is  the  primary  vol- 
tage drop  due  to  the  secondary  coil  resistance  (the  ratio  of  trans- 
formation is  taken  as  unity  for  convenience)  ; OK  is  the 
primary  voltage  drop  due  to  the  resistance  of  the  load  ; HO 


500 


ALTERNATING  CURRENTS 


is  the  mutually  induced  voltage.  In  this  case,  for  ease  in  con- 
struction, the  secondary  current  is  scaled  so  that  it  equals 
OC  in  length.  It  will  coincide  with  that  line  because  there 
is  no  appreciable  reactance  present. 

Now  if  positive  or  negative  reactance  is  introduced  so  that 
the  apex  of  the  right-angled  triangle  of  voltages  takes  any  posi- 
tion on  the  locus  OMON  as  Q' , Q" , or  Q'",  the  ends  of  the  lines 
representing  the  current  vectors  for  the  various  conditions  will 
also  fall  on  the  locus  OMCN ; for  it  was  proved  in  an  earlier 
chapter  that  when  a circuit  contains  variable  reactance  and  con- 
stant resistance,  the  locus  of  the  current  is  a circle  having  a di- 
ameter lying  in  the  line  of  the  impressed  voltage  and  equal  to  the 

impressed  voltage  divided  by  the  resistance,  or  — . This  is 

R 

the  maximum  current  and  is  represented  by  00  in  Fig.  281. 
The  diameter  of  the  circular  locus  is  equal  to  this  current. 
The  secondary  current  for  a certain  positive  reactance  is  then 
Q'  0 and  the  primary  current  for  the  same  reactance  is  AQ' , 
where  AO  is  the  exciting  current.  The  line  CH'K'  being 
drawn  perpendicular  to  CQ'  and  K' K\  being  parallel  thereto, 
the  mutually  induced  voltage,  if  there  is  no  magnetic  leakage 
in  the  coils,  but  when  the  reactance  voltage  of  the  load  is  CQ', 
is  H'  0,  and  the  primary  and  secondary  coil  resistance  drops  are 
respectively  J\Q'  = H'C  and  K\J\  = K'H' . The  total  ac- 
tive primary  voltage  is  OQ'  = K'  0 + OK' v and  the  secondary 
terminal  voltage  is  K'  -f).  It  will  be  noted  from  the  construc- 
tion that  as  Q moves  along  its  locus  by  reason  of  a change  of 
reactance,  K moves  around  a circle  having  a diameter  OK,  and 
H describes  a circle  with  a diameter  OH.  The  smaller  circle 
is  evidently  the  locus  of  the  mutually  induced  voltage  HO  and 
of  the  resistance  voltage  drop  HO  of  the  primary  coil,  and  the 
larger  circle  is  the  locus  of  the  resistance  drop  KO  of  the  pri- 
mary and  secondary  coils  combined. 

When  magnetic  leakage  is  present,  the  diagram  for  a non- 
inductive  load  is  as  in  Fig.  277.  In  Fig.  281  the  coil  voltage 
drops  are’  similarly  shown  for  a non-reactive  load  where  mag- 
netic leakage  is  present,  by  the  lines  FXC,  H1FV  J1HV  and 
KXJV  and  the  secondary  current  is  then  on  the  line  Gq  0.  Then 
when  reactance  is  introduced  into  the  load  and  Q moves  on 


* Art.  70,  Case  2. 


MUTUAL  INDUCTION,  TRANSFORMERS 


501 


the  locus  OMCN , the  figure  CGr1KlHl  swings  around  and 
remaining  symmetrical  diminishes  in  size  when  the  current  de- 
creases, as  did  the  line  OK  when  leakage  was  absent.  When 
Q reaches  0,  the  secondary  current  becomes  zero  and  only  the 
exciting  current  remains  in  the  primary  circuit,  for  then  the 
reactive  voltage  equals  the  impressed  voltage  minus  the  small 
component  required  to  drive  the  exciting  current  through  the 
primary  coil.  As  this  requires  infinite  reactance  it  is  equiva- 
lent to  a transformer  with  the  secondary  circuit  open.  The  right- 
hand  semicircle  OQ'  C is  the  current  locus  for  varying  net  in- 
ductance and  OQ"' C for  varying  net  capacity. 

The  broken  arc  of  a circle  M2  0N2  is  of  the  circular  locus  that 
would  be  formed  if  the  secondary  currents  for  varying  induct- 
ance were  drawn  outward  from  0 instead  of  inward. 

Prob.  1.  Given  a 50-kilowatt  60-cycle  transformer  transform- 
ing from  2200  to  220  volts,  in  which  the  primary  and  secondary 
coil  resistances  are  .8  and  .008  ohm  respectively,  the  core 
losses  100  watts  (assumed  constant),  the  exciting  current  .3  of 
an  ampere,  and  the  magnetic  leakage  negligible.  When  the 
load  resistance  is  maintained  such  that  the  output  is  equal  to 
50  kilowatts,  the  primary  voltage  being  2200  volts,  what  are 
the  primary  and  secondary  currents  for  power  factors  of  1,  .8,  .5, 
and  .2,  for  both  positive  and  negative  secondary  angles  of  lag? 

133.  The  Use  of  Equivalent  Impedances  in  Solving  Transformer 
Problems.  Vector  Formulas. — It  is  possible  to  substitute  im- 
pedance coils  for  the  transformer  which  will  duplicate  the  trans- 
former diagrams  in  Figs.  277  to  281.  Thus,  in  Fig.  282,  Zv  Zv 


M N P 


Fig.  282.  — Combination  of  Impedances  Equivalent  to  the  Impedances  of  a 
Transformer. 

and  Z represent  three  sets  of  impedances  connected  in  series 
which  are  equal  respectively  to  the  impedances  of  the  primary 
coil,  Zx  = V + Au2,  the  secondary  coil,  Z2  — V Ii22  + X2L2, 
and  the  load,  Z — Vi?2  + A2,  of  a transformer  after  its  secon- 


502 


ALTERNATING  CURRENTS 


dary  quantities  have  been  reduced  to  primary  equivalents.  A 
branch,  which  carries  a current  equal  to  the  exciting  current, 
is  connected  across  the  main  circuit  at  MT.  The  arm  Rc  is  of 
such  resistance  that  an  active  current  flows  which  when  multi' 
plied  by  the  impressed  voltage,  Ev  gives  a product  equal  to  the 
no  load  losses  (see  DO , Fig.  277);  while  the  arm  Xm  is  such  a 
reactance  as  will  permit  the  flow  of  the  proper  magnetizing 
element  {AD,  Fig.  277)  of  the  exciting  current;  the  two  com- 
bine to  give  the  total  exciting  current  I^{AO,  Fig.  277). 
Under  this  arrangement  the  impressed  voltage  and  current 
of  the  series  of  impedances  are  Ev  1^,  the  load  voltage  and 
current  are  Ev  I2;  and  the  angles  of  lag  are  0V  0V  These 
and  component  voltages  and  currents  are  as  shown  in  Figs. 
277  to  281;  the  particular  figure  and  the  character  of  the  loci 
which  apply  depending  upon  the  dimensions  and  character  of 
the  various  impedances. 

In  order  that  the  combination  may  be  a complete  reproduc- 
tion of  conditions  of  a transformer  the  exciting  current  path 
should  shunt  the  second  impedance  and  load  at  XS.  The 
current  flowing  in  it  would  then  be  proportional  to  the  voltage 
induced  in  the  primary  coil  by  the  mutual  flux  {SO,  Fig.  277 
and  other  figures)  and  its  vector  would  make  a constant  angle 
therewith,  instead  of  being  proportional  to  and  at  a fixed  angle 
with  the  impressed  voltage. 

The  first  arrangement  is  sufficiently  accurate  for  ordinary 
calculations  and  is  a less  complicated  arrangement.  It  is 
noticed  that  by  the  substitution  of  impedances  the  solution 
of  transformer  problems  becomes  in  principle  equivalent  to  the 
simple  solutions  of  circuits  dealt  with  quite  extensively  in  an 
earlier  chapter.* 

The  vector  equations  necessary  for  determining  the  most  im- 
portant characteristics  of  the  transformer  may  be  readily  writ- 
ten from  Fig.  282,  thus:  Considering  the  first  arrangement  of 
impedances  illustrated  in  Fig.  282,  the  total  impedance  offered 
to  the  impressed  voltage  bv  the  main  circuit  J\INPQ  is 

Zq  = Z1  + Zy  -(-  Z , 

Z0  = {Rt  -(-  R2  + It ) ■+■  j {X1L  + A2i  ± X)  ; 
where  Zv  Rv  and  X1L  are  the  impedance,  resistance,  and  leak- 


* Al  t.  86. 


MUTUAL  INDUCTION,  TRANSFORMERS 


503 


age  reactance  respectively  of  the  first  coil,  Zv  Rv  and  X2i  of 
the  second  coil,  and  Z,  R,  and  X of  the  load;  X is  negative 
when  capacity  reactance  predominates  in  the  load.  Let  the  first 
term  on  the  light  equal  Rt  and  the  second  equal  ±jXt.  The 
admittance  of  the  circuit  is  then 


Y — ^ 

0 R ? + X,2 


L 3 


R?  + X? 


and  the  current  flowing  through  it,  when  the  voltage  E1  is 
impressed,  is 


R,Rt  , E,Xt  , 
/.’r+x;2  J R?  + X? 


The  two  terms  in  the  right-hand  member  of  this  equation  rep- 
resent respectively  the  active  and  reactive  components  of  I±. 
This  current  represents  the  secondary  current  in  a transformer 
having  a unity  ratio  of  transformation  and  the  resistances  and 
reactances  of  the  windings  are  as  denoted  in  Fig.  282,  the  vector  . 
angle  being  measured  from  the  impressed  voltage  Ev 
The  total  primary  current  flowing  is 


I\  J\  T I ix  ! Ifx  — Ic  3 Ym, 

where  _Z^,  Ic , and  Im  are  the  total  exciting  current  and  its 
active  and  reactive  components  respectively  when  referred  to 
the  impressed  voltage  E1  as  the  initial  vector  direction.  Then, 


calling 


X\Rt  _ j 

R?  + X?  R‘ 


and 


x,xt  r 

R?  + X2 


there 


results 


I\  — (Ir,  + Ic)  —j(Im  ± I,t). 


The  angle  of  lag  between  Ev  the  impressed  voltage,  and  Iv  the 
total  current  that  would  flow  in  the  primary  winding  of  the 
equivalent  transformer,  is 


tan  1 


Im  ± I.Xt 

I Rt  d”  Ic 


The  total  voltage  across  the  second  coil  and  load,  which  is 
equal  and  opposite  to  the  total  mutually  induced  primary  and 
secondary  coil  voltages  in  the  equivalent  transformer  of  unity 
ratio  of  transformation,  is 


Ej1  = I1,Z2  + 7/Z 

— I\(H  2 + H)  I J Ii'(XiL  ± X). 


504 


ALTERNATING  CURRENTS 


The  voltage  across  the  load  is 

E2  = I1'Z=I1'R±jI1'X. , 
and  the  angle  of  lag  of  1^  with  respect  to  E2  is 

#2  = tan-1-^p, 

which  is  the  secondary  angle  of  lag  in  the  equivalent  trans- 
former. 

The  power  input  is  -Eqij  cos  6l ; the  power  output,  E2I2  cos  02  ; 
the  excitation  losses  are  I^RC\  the  coil  resistance  losses  due 
to  //  are  T1,2K1  and  I1'2E2.  The  regulation  for  any  load,  neg- 

lecting  the  small  drop  due  to  the  exciting  current,  is  — — 2, 

-®i 

where  E2  is  the  voltage  across  an  impedance  which  is  equivalent 
to  that  load.  The  commercial  efficiency,  or  the  ratio  of  the 
output  to  the  input  for  any  load,  can  be  readily  calculated  from 
the  data  obtained  by  the  formulas  above. 

If  any  difficulty  attaches  to  a clear  understanding  of  the 
formulas  just  given,  it  is  well  to  lay  out  impedance  and  admit- 
tance triangles  for  the  equivalent  impedances  and  admittances 
(Fig.  283)  exactly  as  is  done  in  Chapter  VI,  though  reference 

to  Figs.  277  and  281  may 
serve  as  well,  as  they  repre- 
sent voltage  and  current 
diagrams  of  the  same  general 
character.  After  the  data 
required  for  a transformer 
Fig.  283.  — Impedance  Diagram  obtained  from  are  obtained  by  the  use  of 
Equivalent  Transformer  Impedances.  equivalent  impedances, 

which  requires  the  reduction  of  the  secondary  quantities  to 
primary  equivalents,  the  secondary  currents  and  voltages  may 
be  stated  in  their  correct  values  by  multiplying  and  dividing 
respectively  by  the  ratio  of  transformation  s. 

Prob.  1.  A 500-kilowatt,  60-cycle,  5500-volt  (nominal  pri- 
mary voltage)  transformer  having  a ratio  of  transformation  of 
20  to  1 develops  its  full  capacity  with  a voltage  of  230  volts  at 
the  secondary  terminals.  At  full  load  the  copper  losses  in  the 
primary  and  secondary  coils  are  for  each  ^ per  cent  of  the  trans- 
former’s output.  The  drop  in  voltage  at  full  load  due  to  mag- 


MUTUAL  INDUCTION,  TRANSFORMERS 


505 


netic  leakage  is  1 per  cent  of  the  no  load  voltage.  The  iron 
losses  are  1 per  cent  of  the  full  load  output,  and  the  exciting 
current  is  2 amperes.  The  load  reactance  causes  the  current  to 
lead  the  secondary  terminal  voltage  by  an  angle  of  30°.  Find 
by  the  method  given  in  this  article  the  primary  voltage,  pri- 
mary current,  and  primary  angle  of  lag,  and  the  secondary  cur- 
rent. Find,  also,  the  primary  current,  and  the  secondary  vol- 
tage and  current  when  the  rated  voltage  (5500  volts)  is  impressed 
on  the  primary  winding,  the  power  factor  of  the  load  being  as 
before.  Give  the  regulation  and  efficiency  in  each  case. 
Answer  the  same  questions  when  the  load  is  reduced  to  one 
half  rated  capacity  (250  kilowatts),  assuming  that  the  iron  losses 
and  exciting  current  do  not  change. 

134.  The  Effect  of  Harmonics  in  the  Waves  of  Voltage  and 
Current  upon  the  Operation  of  a Transformer.  — If  the  primary 
voltage  impressed  on  a constant  voltage  transformer  in  a single 
phase  circuit  contains  harmonics,  the  harmonics  are  transmitted 
into  the  voltage  of  the  secondary  circuit,  since  the  form  of  the 
wave  of  magnetism  in  the  core  is  determined  by  the  form  of 
the  impressed  voltage  wave  after  the  small  part  that  is  absorbed 
in  copper  losses  has  been  deducted,  and  the  mutually  induced 

secondary  voltage  is  at  each  instant  proportional  to  — 

Ctb 

If  the  secondary  circuit  contains  considerable  capacity  re- 
actance and  comparatively  small  resistance,  the  higher  harmonics 
of  the  secondary  voltage  cause  exaggerated  current  harmonics.* 
The  magneto-motive  forces  of  these  secondary  current  harmonics 
would  affect  the  magnetic  flux  of  the  core  and  they  must  be  off- 
set by  opposing  magneto-motive  forces  in  the  primary  winding, 
which  results  in  harmonics  of  current  flowing  in  the  primary 
circuit  that  are  equivalent  to  those  in  the  secondary  circuit. 

Thus,  irregular  voltages  and  currents  in  the  circuits  to  which 
a transformer  is  attached  act  very  much  as  would  be  the  case  if 
the  transformer  was  replaced  by  the  equivalent  network  dis- 
cussed in  the  previous  article. 

As  the  exciting  current  of  a transformer  is  quite  irregular 
in  form,  it  contains  harmonics  of  relatively  large  amplitude. 
Therefore,  in  a circuit  containing  a large  number  of  unloaded 
or  lightly  loaded  transformers,  conditions  may  arise  under 


* Art.  69. 


,506 


ALTERNATING  CURRENTS 


which  the  resultant  flow  of  irregular  current  may  prove 
troublesome.  The  principal  harmonic  in  the  exciting  current 
of  higher  frequency  than  the  fundamental  when  the  impressed 
voltage  is  sinusoidal  is  the  third,  although  the  fifth  also  may 
be  observed.  The  conditions  may  be  understood  from  Fig. 
263,  which  shows  a hysteresis  and  eddy  current  cycle  and  the 
current  wave  required  to  produce  a sinusoidal  magnetic  flux. 
The  quadrature  component  of  current  is  peaked  as  a result  of 
the  increased  reluctance  caused  by  partial  saturation  of  the 
iron  toward  the  top  of  the  flux  wave,  which  occurs  at  even 
the  relatively  low  magnetic  densities  used  in  transformers. 
This  symmetrical  peaking  of  the  quadrature  current  compo- 
nent indicates  a prominent  third  harmonic  therein. 

No  matter  what  the  primary  voltage  wave  may  be  like,  the 
secondary  voltage  wave  must  have  practically  the  same  form, 
the  exciting  current  and  magnetization  curves  varying  to  meet 
this  requirement.  When  the  voltage  curve  is  peaked,  the 
magnetization  curve  is  flattened  and  does  not  reach  so  high  a 
maximum  as  for  an  equivalent  sine  curve ; and  conversely, 
when  the  voltage  curve  is  flat,  the  magnetization  curve  will  be 
peaked,  as  has  been  hereinafter  proved  in  Art.  157.  Since  the 
iron  losses  of  a transformer  depend  upon  the  maximum  value  of 
the  magnetic  flux  in  the  core,  it  is  to  be  expected  that  a peaked 
voltage  curve  will  cause  minimum  iron  losses.  Roessler  has 
shown  that  the  iron  losses  of  a transformer  depend  upon  the  ratio 
of  the  average  value  of  the  voltage  wave  to  its  effective  value, 
i.e.  upon  the  Form  Factor.  Steinmetz  cites  a case  where  the 
losses  in  a 200-kilowatt  transformer  were  13  per  cent  less  when 
worked  on  a peaked  curve  than  when  a sine  curve  was  im- 
pressed. The  change  in  full  load  efficiency  is,  however,  very 
slight  in  ordinary  commercial  practice,  although  the  “all  da}'” 
efficiency  may  be  considerably  affected  by  the  form  of  the  vol- 
tage wave  impressed  on  a transformer  which  runs  for  any  con- 
siderable part  of  the  day  very  lightly  loaded.  Generally  speak- 
ing, constant  voltage  transformers  have  the  best  efficiencies 
when  operated  on  circuits  having  peaked  voltage  waves.  Ex- 
perience shows  that  alternating  current  arc  lights  give  the  best 
results  when  worked  on  a flat-topped  curve,  while  a sine  curve 
is  most  convenient  for  the  operation  of  synchronous  motors  and 
gives  the  best  efficiency  in  the  operation  of  induction  motors. 


MUTUAL  INDUCTION,  TRANSFORMERS 


507 


The  last  statement  is  shown  later  to  be  theoretically  correct.* 
Since  a transformer  may  be  required  to  supply  any  one  of  these 
kinds  of  loads,  an  approximate  sine  wave  probably  gives  the 
best  general  results. 

135.  Effects  of  Changes  of  Frequency  and  Voltage.  — The  for- 
mulas giving  the  voltages  developed  in  the  coils  of  an  unloaded 
transformer, 


^ = 2 


V2  7 T<&fn 


1 08 


1 and  -E"2  — -%J2V  f 


show  that  if  the  maximum  value  of  the  magnetic  flux  in  the 
core  of  a transformer  is  fixed,  then  for  a given  impressed  vol- 
tage the  number  of  turns  in  the  coils  must  vary  inversely  with 
the  frequency.  To  design  transformers  for  all  frequencies  with 
a fixed  value  of  the  maximum  magnetic  flux  is  not  an  economi- 
cal plan,  however,  since  the  core  loss  per  cubic  centimeter  of 
iron  depends  upon  the  frequency.  If  a certain  core  loss  is  de- 
termined upon  as  being  that  which  will  give  the  most  satis- 
factory general  results  in  the  case  of  a transformer  of  given 
size  and  for  a given  duty,  then  the  magnetic  density  in  the 
core  must  be  designed  to  be  smaller  as  the  frequency  is  made 
larger.  This  may  be  seen  from  the  fact  that  the  eddy  current 
loss  varies  as  the  square  of  the  frequency,  and  the  hysteresis 
loss  as  the  first  power  of  the  frequency,  other  things  being 
equal.  Since  the  former  is  not  uncommonly  between  one 
tenth  and  one  fourth  of  the  latter,  this  makes  the  total  core 
losses  vary  somewhere  between  the  first  and  second  powers  of 
the  frequency,  if  the  maximum  magnetic  density  is  designed  to 
be  the  same  for  all  transformers.  The  eddy  current  loss  also 
varies  as  the  square  of  the  maximum  magnetic  density,  and  hys- 
teresis loss  varies  with  some  degree  of  approximation  within 
the  limits  of  transformer  practice,  as  the  1.6th  power  of  the 
maximum  density.  Consequently,  the  total  losses  vary  as 
some  power  of  the  maximum  density  between  the  1.6th  and 
the  second.  It  is  thus  shown  that  the  core  losses  will  not  be 
greatly  changed  if  the  maximum  magnetic  density,  within 
reasonable  limits,  varies  inversely  with  the  frequency.  Now 
it  is  evident  from  the  formula  that,  if  the  turns  and  voltage  re- 
main unchanged,  when  the  frequency  changes,  the  maximum  flux 


* Art.  198. 


t Art.  122. 


508 


ALTERNATING  CURRENTS 


will  change  inversely ; hence  a transformer  should  give  nearly 
the  same  efficiency  for  currents  of  different  frequencies  reason- 
ably near  that  for  which  it  was  designed,  only  a small  rate  of 
increase  of  the  iron  losses  occurring  as  the  frequency  decreases  ; 
and  the  number  of  turns  in  the  coils  may  be  nearly  the  same 
in  a series  of  transformers  of  equal  capacities  designed  for  the 
same  voltages  but  different  frequencies.  This  statement  is 
made  under  the  supposition  that  the  lower  frequencies  do  not 
call  for  a maximum  magnetic  flux  which  exceeds  the  saturation 
point  of  the  magnetic  core. 

Changing  the  frequency  alters  the  quadrature  component  of 
the  exciting  current  in  a proportion  which  depends  upon  the 
saturation  of  the  core,  and  transformers  built  of  poor  material, 
or  which  operate  with  their  cores  near  the  point  of  saturation 
when  used  on  normal  frequency,  may  give  unsatisfactory  service 
and  overheat  on  lower  frequencies  than  their  normal,  on  account 
of  an  excessive  magnetic  density  and  an  excessive  exciting 
current.  Transformers  under  these  circumstances  may  have 
poor  regulation  due  to  large  copper  drop  and  increased  magnetic 
leakage.  This  is  seen  from  the  voltage  formula,  where  d>/ 
must  be  constant  while  is  constant.  It  will  be  seen  also  from 
the  formula  that  raising  Ex  above  the  rated  value  tends  to  cause 
a greater  maximum  flux  <f>,  so  that  even  if  the  insulation  will 
stand  an  increased  voltage,  the  increase  cannot  go  beyond  the 
point  where  the  core  becomes  saturated,  without  causing  ex- 
cessive core  losses  and  an  excessive  flow  of  exciting  current. 

Since  the  rates  of  variation  of  the  hysteresis  and  eddy  cur- 
rent losses  with  the  frequency  and  with  the  reciprocal  of  the 
maximum  magnetic  density  are  not  exactly  the  same,  it  is  to 
be  expected  that  the  efficiency  of  a transformer,  so  far  as  it  is 
affected  by  the  iron  losses,  will  vary  to  some  degree  with  the 
frequency. 

Since  the  flux  density  varies  inversely  as  the  frequency  of 
the  impressed  voltage  in  any  particular  transformer,  as  is  shown, 
by  the  foregoing  formula,  and  the  eddy  current  loss  is  propor- 
tional on  the  one  hand  to  the  square  of  the  frequency  and  on 
the  other  hand  to  the  square  of  the  flux  density,  a variation  of 
the  frequency  leaves  the  eddy  current  loss  substantially  unaf- 
fected, provided  the  paths  for  the  flow  of  the  eddy  currents  do 
not  change.  In  the  case  of  the  hysteresis  loss,  different  consid- 


MUTUAL  INDUCTION,  TRANSFORMERS 


509 


erations  arise,  for  this  loss  is  proportional  to  /T1-6,  while  <I>  in 
any  particular  transformer  is  proportional  to  1//.  The  conse- 
quence is  that  the  hysteresis  loss  in  a particular  transformer, 
when  the  frequency  is  varied,  changes  proportionally  with 
l//-6.  In  first  class  modern  transformers,  the  eddy  current 
loss  is  usually  much  smaller  than  the  hysteresis  loss,  and  the 
efficiency  of  a particular  transformer  increases  appreciably  as 
the  frequency  of  the  impressed  voltage  is  increased. 

Transformers  of  equal  outputs  but  for  different  frequencies 
may  be  built  with  windings  having  like  numbers  of  turns  but 
with  the  cross  sections  of  the  cores  inversely  proportional  to  the 
frequency.  The  magnetic  density  would  then  be  the  same  in 
all  the  transformers.  This  is  unsatisfactory,  however,  since  it 
makes  low  frequency  transformers  large  and  expensive ; and 
since  the  weight  of  iron  in  a core  must  vary  more  rapidly  than 
the  cross  section,  this  method  of  construction  also  causes  com- 
paratively large  core  losses  in  low  frequency  transformers  and 
gives  them  comparatively  low  efficiency. 

The  conclusion  is  therefore  derived  that  the  core  and  wind- 
ings of  a well-designed  and  well-constructed  transformer  will 
be  satisfactory  in  performance  for  small  variations  in  frequency, 
but  that  transformers  designed  for  high  and  low  frequencies 
respectively  should  be  made  with  such  numbers  of  turns, 
weights  of  iron,  and  magnetic  densities  as  will  give  in  each 
case  the  efficiency  and  regulation  sought.  With  the  modern 
high  grade  alloy  steels  the  hysteresis  is  so  much  less  than  in 
common  steels  and  irons  that  it  is  possible  to  carry  the  magnetic 
density  well  toward  the  point  of  saturation  for  all  frequencies, 
and  this  is  now  standard  practice. 

The  modern  standard  frequencies  in  this  country  are  60 
cycles  per  second  for  lighting  and  small  power  purposes,  25 
cycles  per  second  for  alternating  current  railroad  and  large 
power  units,  and  sometimes  40  cycles  per  second  for  combined 
lighting  and  power  circuits. 

136.  Transformer  Iron  and  Steel.  — Early  experience  showed 
that  the  core  losses  in  some  transformers  increased  to  a very  con- 
siderable extent  during  the  first  few  months  of  operation.  The 
increase  in  losses  was  found  to  be  due  to  increased  hysteresis  loss 
per  cycle,  and  was  originally  ascribed  by  Ewing  to  magnetic 
fatigue  of  the  iron.  Later  it  was  conclusively  proved  by 


510 


ALTERNATING  CURRENTS 


experiments  and  the  records  of  transformer  manufacturers  to 
be  caused  by  the  continuous  condition  of  heightened  tempera- 
ture at  which  the  iron  is  operated.  This  effect  of  Ageing  seems 
to  have  the  greatest  effect  upon  poor  qualities  of  iron,  hastily 
and  imperfectly  annealed,  and  the  least  effect  upon  the  best 
grades  of  iron  which  have  been  annealed  with  great  care.  The 
conditions  under  which  the  annealing  of  transformer  iron  is 
performed,  especially  with  reference  to  temperature  and  dura- 
tion of  the  process,  have  much  to  do  with  the  extent  of  the 
ageing  effect,  and  by  proper  annealing  it  can  be  rendered  very 
small  in  cores  made  of  proper  qualities  of  iron.  The  iron 
formerly  and  now  much  used  for  transformer  cores  is  a very 
mild  steel  made  by  either  the  bessemer  or  open-hearth  process, 
though  puddled  iron  sheets  were  formerly  used  to  some  extent. 
But  the  magnetic  material  most  approved  at  the  present  day  is 
an  alloy  of  silicon  with  mild  steel.  This  is  steel  containing 
from  2 1 to  4 or  more  per  cent  of  silicon,  called  silicon  steel. 
Such  steels  are  harder  and  more  difficult  to  make  and  work,  so 
that  they  are  more  expensive  than  ordinary  soft  steel.  But 
these  steels  are  apparently  little  affected  by  ageing — which 
sometimes  increases  the  hysteresis  losses  in  ordinary  steels  as 
much  as  100  and  150  per  cent  after  the  metal  has  been  in 
service  in  a transformer  core  for  a few  months.  Further, 
these  new  steels  have  a lower  hysteretic  constant,  and  trans- 
former cores  made  of  some  of  them  waste  from  35  to  45  per 
cent  less  power  in  hysteresis  than  do  the  ordinary  steels.  They 
also  possess  higher  specific  resistances,  and  this  reduces  eddy 
current  losses.  Thus,  for  the  same  efficiency  and  duty  a trans- 
former can  be  built  having  a core  of  the  best  transformer  silicon 
steel  at  a much  less  weight  per  kilowatt  capacity,  than  when  any 
one  of  most  of  the  standard  soft  steels  is  used.  This  is  be- 
cause the  magnetic  density  can  be  increased  from  45  to  65  per 
cent,  without  increasing  the  loss  due  to  hysteresis,  which  re- 
sults in  a consequent  reduction  in  the  cross  section  of  the  core 
and  a resultant  lessening  of  the  length  of  wire  per  turn.*  A 
fair  value  for  the  hysteretic  constant  for  good  soft  transformer 
steel  is  .0021  and  for  silicon  steel  .00094  in  terms  of  watts  per 
pound  and  per  cycle.*  In  Fig.  284  the  curves  A and  A'  are 
plotted  from  the  hysteresis  equation  f and  show  the  hysteresis 

* Trans.  A.  I.  E.  E.,  Yol.  28,  p.  439;  also  Vol.  30.  t Art.  106. 


MUTUAL  INDUCTION,  TRANSFORMERS 


511 


loss  in  watts  per  cubic  centimeter  and  per  pound  of  metal  when 
the  frequency  of  magnetic  cycles  is  100  per  second  and  the 


hysteresis  constant  is  rj  = .0021.  Curve  A is  plotted  for  max- 
imum flux  densities  from  0 to  5500  lines  of  force  per  square 


512 


' ALTERNATING  CURRENTS 


centimeter,  and  curve  A!  is  plotted  for  maximum  flux  densities 
from  5000  to  10,500;  while  the  curves  B , B ',  and  B " are  plotted 
for  7)  = .00094,  the  maximum  flux  densities  being  0 to  5500 
lines  of  force  per  square  centimeter  for  curve  B , 5000  to  10,500 
lines  of  force  per  square  centimeter  for  curve  B\  and  10,000  to 
15,500  for  curve  B" . 

It  must  be  remembered  that  the  character  of  steels,  especially 
of  silicon  or  other  alloyed  steels,  is  apt  to  be  quite  variable. 
Therefore,  in  designing  magnetic  apparatus,  the  exact  kind  of 
material  to  be  used  should  be  tested  and  its  qualities  as  thus 
determined  used  in  making  calculations.  As  seen  from  the 
hysteresis  equation  just  referred  to,  it  is  evident  that  the 
curves  of  Fig.  284  may  be  used  for  any  other  hysteresis 
constant  and  frequency  by  multiplying  the  vertical  scale  of 

losses  by  the  ratio  where  t)f  is  the  product  of  the  hysteresis 
VJ 

constant  and  frequency  taken  in  calculating  the  curves  referred 
to,  and  7]'f  is  the  product  of  the  values  of  those  quantities  for 
which  the  data  are  sought. 

The  eddy  current  losses  are  dependent  both  upon  the  thick- 
ness of  the  sheets  and  the  specific  electrical  resistance  of  the 
steel.  The  thickness  of  laminations  quite  commonly  used  for 
60-cycle  transformers  is  from  14  to  15  mils,  or  more  exactly  No. 
29  sheet-iron  gauge,  which  has  a thickness  of  14.1  mils,  while 
for  25  cycles  No.  26  gauge  sheets,  having  a thickness  of  18.7  mils, 
do  not  permit  the  eddy  current  loss  to  exceed  economical  limits. 
The  specific  resistance  of  iron  varies  quite  widely,  and  this  is 
especially  true  of  the  alloy  steels,  which  may  have  a much 
greater  resistance  than  the  soft  steels  and  irons.  In  Fig.  285 
curves  A , B , and  O are  the  calculated  eddy  current  losses  in 
plates  of  10,  15,  and  20  mils  thickness  respectively  for  various 
maximum  magnetic  densities,  a frequency  of  100  periods  per 
second,  and  a specific  resistance  of  1 x 10-5,  which  is  a fair 
value  for  good  transformer  steel.*  The  curve  shown  in  three 
sections,  Z),  D\  D'\  is  for  a plate  15  mils  in  thickness  and  having 
a specific  resistance  three  times  as  great  as  for  the  first  curves. 
It  represents  some  of  the  harder  alloys.  The  curve  can  readily 
be  used  for  other  frequencies  and  thicknesses  of  laminations,  as 
will  be  seen  by  an  inspection  of  the  eddy  current  formula. f 


* Art.  111. 


t Ibid. 


MUTUAL  INDUCTION,  TRANSFORMERS 


513 


The  total  core  loss  in  transformers  varies  more  or  less,  de- 
pending upon  the  characteristics  desired,  being  from  about  1.1  to 

001  = / "WO  -no  S3d  S11VM 


3 per  cent  in  small  transformers,  such  as  those  under  5 kilowatts 
capacity,  and  decreasing  to  about  | of  a per  cent  in  those  of  25  kilo- 

2* 


514 


ALTERNATING  CURRENTS 


1GOOO 


14000 


12000 


£ 10000 


8000 


C5  6000 


4000 


2000 


rB 

c 

-A 

12 


Fig.  286. - 


2 4 6 8 10 

MAGNETIZING  FORCE,  H 

-Curves  of  Magnetization  of  Transformer 
Steels. 


14 


watts  capacity,  to  § 
of  a per  cent  in  those 
of  50  kilowatts  ca- 
pacity, and  to  a 
smaller  percentage  as 
the  capacity  still  fur- 
ther increases.  The 
eddy  currents  seldom 
comprise  over  10  per 
cent  of  the  total  core 
losses. 

As  the  best  grades 
of  low  carbon  steels 
and  wrought  irons 
are  used  for  trans- 
former cores,  the 
ratio  between  the 


magnetic 


maximum 

density  <£  and  the 
field  strength  H is  high.  Curve  B*  in  Fig.  286  is  from  a good 
grade  of  transformer  steel,  and  curve  A represents  a soft  steel  of 
poorer  grade.  The 
silicon  steels  do  not 
ordinarily  have 
quite  such  high 
curves  as  the  best 
transformer  carbon 
steels.  The  rela- 
tion  between 
curves  B and  C 
fairly  indicates  the 
difference.  The 
permeability  of  B 
in  Fig.  286  is  shown 
by  B in  Fig.  287. 

Curve  A is  a poorer 
grade  of  soft  iron 

and  C represents  0 '-’00°  4000  6000  8000  10000 

t.  . - MAGNETIC  DENSITY  = <#> 

tll6  permeability  of  Fig.  287.  — Curve  of  Permeability  of  Transformer  Steels, 

* Smithsonian  Tables,  pp.  274  and  275. 


12000  14000 


MUTUAL  INDUCTION,  TRANSFORMERS 


515 


a good  grade  of  silicon  steel.  The  silicon  steels  take  somewhat 
more  exciting  current  than  the  best  grades  of  soft  iron  and  soft 
carbon  steels,  on  account  of  their  lower  permeability. 

The  maximum  magnetic  densities  to  which  transformer  cores 
are  worked  is  dependent  somewhat  upon  the  frequencies  for 
which  the  transformers  are  designed.  For  60  cycles  the  den- 
sities vary  from  6000  to  10,000  lines  of  force  per  square  centi- 
meter, or  slightly  higher,  with  a tendency  toward  10,000  when 
the  highest  grade  transformer  core  metal  is  used.  For  25  cycles 
they  vary  from  nine  or  ten  to  twelve  or  more  thousand  lines  of 
force  per  square  centimeter.  The  usual  silicon  steels  cannot 
be  run  too  high  in  magnetic  density  or  the  exciting  current 
will  be  larger  than  desired.  In  silicon  steels  a maximum  mag- 
netic density  of  10,000  lines  of  force  per  square  centimeter  is 
apt  to  be  about  as  near  the  point  of  saturation  as  it  is  wise 
to  go,  while  economy  of  material  dictates  as  high  a density  as 
possible,  so  that  in  modern  transformers,  using  the  new  metals, 
the  density  will  not  ordinarily  be  far  from  that  figure.  With 
ordinary  steels  the  greater  iron  losses  tend  to  make  lower  den- 
sities more  desirable. 

In  addition  to  silicon  there  are  other  alloying  materials  used 
with  iron  for  the  purpose  of  reducing  the  hysteresis  coefficient 
below  figures  common  in  ordinary  soft  carbon  steels.  Among 
these  are  nickel,*  tin,  and  arsenic,  f The  new  pure  irons  re- 
cently produced  have  also  unusually  good  magnetic  qualities. 
The  state  of  the  art  in  the  manufacture  of  iron  and  steel  is 
rapidly  advancing,  and  it  is  probable  that  present  practice  will 
be  materially  modified  as  a result. 

137.  Efficiencies  of  Transformers.  — The  average  working  effi- 
ciency of  a transformer  is  by  no  means  equal  to  its  full  load 
efficiency.  In  the  case  of  dynamos  placed  in  a central  station 
it  is  usual  to  divide  the  generating  units  so  that  the  plant  op- 
erating during  any  part  of  the  day  will  be  well  loaded.  In 
the  same  way  the  capacities  of  stationary  motors  may  ordinarily 
he  chosen  so  that  the  motors  seldom  operate  at  very  small  par- 
tial loads.  The  way  that  transformers  are  usually  operated, 
however,  makes  it  quite  difficult  to  keep  a uniform  load  on 
them,  and,  in  fact,  for  a considerable  portion  of  the  day  they 
may  have  their  secondary  circuits  open.  When  this  is  the  case, 

t Electromet.  Ind.,  Vol.  VII. 


* Art.  109. 


516 


ALTERNATING  CURRENTS 


the  iron  losses  of  transformers  are  of  much  greater  influence  on 
their  All-day  efficiency  than  the  copper  losses,  and  it  is  desir- 
able to  reduce  the  iron  losses  to  a minimum.  For  instance,  sup- 
pose a transformer  of  5000  watts  output  at  normal  full  load 
has  iron  losses  of  125  watts,  and  copper  losses  at  full  load  of 
100  watts.  Then  the  full  load  efficiency  of  the  transformer 
is  95.7  per  cent,  the  half  load  efficiency  is  94.3  per  cent,  and 
the  quarter  load  efficiency  is  90.4  per  cent.  Supposing  that 
the  daily  output  of  the  transformer  is  equivalent  to  25,000 
watts  for  one  hour  (25,000  watt-hours),  obtained  by  full  load 
operation  for  five  hours  and  open-circuit  operation  for  the  re- 
maining 19  hours,  then  the  losses  in  the  iron  core  are  equivalent 
to  3000  watts  for  one  hour,  and  in  the  copper  to  500  watts  for 
one  hour  (neglecting  no-load  copper  losses),  or  a total  of  3500 
watts  for  one  hour.  The  all-day  efficiency  is  then  87.7  per  cent. 
To  increase  this  all-day  efficiency,  it  is  evidently  necessary  to 
decrease  the  iron  losses.  To  do  this  for  a fixed  frequency 
requires  a decrease  in  the  amount  of  iron  in  the  magnetic  cir- 
cuit or  a decrease  of  the  maximum  magnetic  density.  Either 
process  calls  for  an  increase  in  the  wundings  and  consequently 
in  the  copper  losses.  Suppose  that  decreasing  the  core  losses  to 
75  watts  makes  it  necessary  to  increase  the  full  load  copper 
losses  to  150  watts  ; then,  other  things  being  equal,  the  efficien- 
cies become,  full  load,  95.7  per  cent;  half  load,  95.7  per  cent; 
quarter  load,  93.7  per  cent;  and  the  all-day  efficiency,  on  the 
assumption  made  above,  is  90.7  per  cent.  There  is  a saving  by 
the  latter  construction  of  950  watt-hours  in  twenty-four  hours, 
and  in  one  year  of  365  days  the  saving  is  nearly  350  kilowatt- 
hours.  If  one  kilowatt-hour  is  worth  4 cents  to  the  central 
station,  the  difference  in  the  amount  of  energy  wasted  each  year 
by  the  two  transformers  has  a value  of  nearly  fourteen  dollars, 
which  is  a substantial  part  of  the  difference  in  the  original  cost 
of  the  two  transformers.  If  the  average  load  of  the  transformer 
were  less  than  that  assumed,  which  is  frequently  the  case  for 
small  transformers  in  practice,  the  iron  losses  would  have  a still 
greater  influence  on  the  all-day  efficiency. 

A still  greater  decrease  in  the  iron  losses  with  its  attendant  in- 
crease of  copper  losses  would  evidently  raise  the  all-day  efficiency 
to  a higher  point.  Here,  however,  is  met  the  question  of  regula- 
tion, which  will  not  satisfactorily  admit  of  too  8'reat  a copper  loss 


MUTUAL  INDUCTION,  TRANSFORMERS 


517 


at  full  load  on  account  of  the  attendant  drop  of  secondary  vol- 
tage, but  this  difficulty  maybe  met  by  increasing  the  cross  section 
of  the  copper.  This  alternative  causes  an  increase  in  the  cost 
of  the  transformer,  but  a transformer  with  small  losses  and  good 
regulation  is  worth  more  to  the  central  station  than  one  with 
large  losses  or  poor  regulation.  The  advantage  of  decreasing 
the  iron  losses,  which  is  thus  shown,  led  Swinburne  some  years 
ago  to  advocate  and  build  transformers  with  a cyclindrical  iron 
wire  core  under  the  windings,  but  without  closed  iron  magnetic 
circuits.  A diagram  of  this  transformer,  called  the  Hedgehog, 
which  was  somewhat  extensively  used  in  England  at  one  time 
is  shown  in  Fig.  288.*  Decreasing  the  amount  of  iron  in  the 
magnetic  circuit  decreases  the  core  losses  but 
at  the  same  time  increases  the  reluctance,  and 
therefore  increases  the  exciting  current.  This 
is  a decided  disadvantage  if  carried  to  excess. 

While  the  true  magnetizing  current  is  in  quad- 
rature with  the  mutually  induced  voltage  of  the 
transformer,  yet  it  does  result  in  a continuous 
I2R  loss  in  the  conductors  leading  to  the  trans- 
former and  in  the  primary  winding  of  the  trans- 
former itself.  It  also  causes  an  extra  demand 
for  current  from  the  dynamos  supplying  the 
circuit,  so  that  extra  generators  may  have  to  be 
operated  at  periods  of  light  load  in  order  to  sup- 
ply the  quadrature  currents.  In  other  words, 
the  power  factor  of  the  system  as  a whole  is  de- 
creased, with  an  accompanying  loss  of  plant  efficiency.  Finally, 
a large  magnetic  reluctance  causes  a considerable  magnetic 
leakage  and  consequent  increase  in  the  secondary  drop  of  a 
transformer,  and  therefore  impairs  its  regulation. 

Without  entering  into  a controversy  regarding  the  exact  point 
at  which  a high  reluctance  in  the  magnetic  circuit  of  a trans- 
former causes  more  disadvantage  than  is  counterbalanced  by 
the  decreased  iron  losses  brought  about  by  decreasing  the 
amount  of  iron,  the  examples  may  serve  to  show  the  necessity 
of  carefully  designing  the  magnetic  circuit  to  fit  the  conditions 
if  they  can  be  predetermined. 

*For  test  on  transformer  of  this  type  see  Trans.  A.  I.  E.  E .,  Yol.  10,  p.  497  ; 
Elec.  World,  Yol.  22,  p.  357. 


Fig.  288. — Hedge- 
hog Transformer. 


518 


ALTERNATING  CURRENTS 


The  commercial  efficiency  of  a transformer  is  equal  to  the 
watts  delivered  to  the  external  secondary  circuit  by  the  trans- 
former divided  by  the  watts  absorbed  by  the  primary  coil.  It 
may  be  written 

Po  P0 

71  — i = « , 

Pi  P,  + L 

where  Px  and  P2  are  the  power  absorbed  and  delivered  respec- 
tively by  the  primary  and  secondary  coils,  and  L represents 
the  total  losses  of  power  in  the  transformer.  The  losses  are 
made  up  of  the  I2R  losses  in  the  primary  and  secondary  coils, 
eddy  currents  in  the  windings  caused  by  leakage  flux,  and 
the  losses  due  to  hysteresis  and  eddy  currents  in  the  core. 
The  I2R  loss  in  the  secondary  winding  is  directly  proportional 
to  the  square  of  the  load  (secondary  output),  assuming  the 
secondary  terminal  voltage  to  be  constant,  while  the  PR  loss 
in  the  primary  winding  is  nearly  proportional  to  the  square 
of  the  load,  though  it  contains  a small  approximately  constant 
term  due  to  the  exciting  current.  The  rated  capacities  of  very 
large  transformers  are  not  uncommonly  expressed  in  kilo- 
volt-amperes instead  of  kilowatts,  since  the  PR  heating  effect 
is  dependent  upon  the  current,  which  depends  not  only  upon 
the  load  but  also  inversely  upon  the  power  factor  when  the  vol- 
tage is  constant;  but  ordinarily  the  rating  of  small  transformers 
is  in  kilowatts  capacity,  the  assumption  being  that  the  power 
factor  for  which  they  are  specified  is  approximately  unity. 
The  efficiency  will  obviously  have  different  values  for  different 
loads  even  when  power  factor  and  primary  impressed  voltage 
are  kept  constant;  and  also  if  the  load  (i.e.  the  output  in  kil- 
owatts) is  kept  constant,  the  efficiency  will  have  different 
values  for  different  power  factors. 

The  hysteresis  and  eddy  current  losses  vary  from  .5  to  1.5 
per  cent  of  the  full  load  capacity  in  transformers  of  500  or 
more  kilovolt-amperes  full  load  rating,  and  the  percentage  in- 
creases as  the  capacity  decreases.  In  a transformer  of  100 
kilovolt-amperes  capacity  the  losses  may  be  only  a little  higher 
than  the  minimum  given  above,  or  they  maybe  as  much  as  one- 
half  or  more  above  the  maximum  figure  given  above.  Intrans- 
formers  of  from  5 to  100  kilovolt-amperes  capacity  of  a given 
make  and  type,  the  iron  losses  usually  increase  gradually  with 
decreasing  rated  capacities  until  at  the  small  size  named  they  are 


MUTUAL  INDUCTION,  TRANSFORMERS 


519 


from  1 to  2.5  per  cent  or  more  of  the  rated  capacity.  In  very 
small  transformers  they  rise  to  from  3 to  5 per  cent  of  the  rated 
capacity.  The  loss  in  watts  per  pound  of  iron  varies  from  less 
than  J a watt  per  pound  for  the  best  grades  of  alloy  steels, 
when  a low  magnetic  density  is  used,  to  as  high  as  2 or  3 watts 
per  pound  where  poorer  grades  of  metal  are  used  and  the 
magnetic  density  used  is  higher. 

The  copper  losses  at  full  load  are  apt  to  be  from  25  to  100  per 
cent  greater  than  the  iron  losses,  though  the  tendency  in 
modern  designing  is  to  keep  these  losses  down,  for  by  increas- 
ing the  weight  of  the  core  and  decreasing  the  number  of  the 
coil  turns,  the  IR  drop  and  the  IX  drop  are  reduced,  and  the 
regulation  is  improved.  The  range  of  variation  in  watts  lost 
per  pound  of  copper  for  full  load  current  in  commercial  trans- 
formers is  in  the  neighborhood  of  from  slightly  less  than  one  to 
six  or  more  watts  per  pound.  For  transformers  to  go  into 
lighting  service  or  for  duties  where  there  are  short  full  load 
peaks,  however,  the  copper  losses  may  to  advantage  be  made 
relatively  still  higher,  with  an  accompanying  reduction  of  the 
iron  losses  ; for  then  the  copper  losses  varying  with  the  load 
do  not  have  so  much  influence  on  the  all-day  efficiency  and 
the  reduction  of  the  fixed  iron  losses  results  in  better  all-day 
efficiency.  This  is  not  always  desirable,  however,  as  there  is 
then  a higher  load  on  the  circuit  during  the  peak  when  all 
power  available  in  the  circuits  may  be  most  valuable.  In  any 
event,  the  copper  losses  must  not  exceed  reasonably  low  values, 
or  the  regulation  of  the  transformer  becomes  too  poor  to 
afford  satisfactory  conditions  for  electric  lighting  where  incan- 
descent lamps  are  used. 

The  United  States  Government  specifies  the  iron  and  cop- 
per losses  shown  in  the  table  on  page  520  for  small  step-down 
lighting  transformers  designed  for  a frequency  of  60  cycles 
per  second,  2200  volts  primary  voltage,  and  ratios  of  transfor- 
mation of  10  and  20  to  one. 

The  losses  shown  are  low,  and  the  relatively  low  iron 
losses  compared  with  the  copper  losses  result  in  the  maximum 
commercial  efficiencies  occurring  at  loads  less  than  the  full 
loads,  with  resulting  high  all-day  efficiencies  compared  with 
those  which  would  be  obtained  were  the  same  total  losses  at  full 
load  equally  divided  between  the  copper  and  the  core. 


520 


ALTERNATING  CURRENTS 


U.  S.  TABLE  OF  IRON  AND  COPPER  LOSSES 


Kva.  Rated 
Capacity 

Ikon  Losses  in 
Watts 

Full  Load  Copper 
Losses 

Total  Fcll  Load 
Losses  in  Pee  Cent 
of  Capacity 

1 

21 

26 

4.70 

2.5 

33 

55 

3.52 

5 

45 

93 

2.76 

10 

82 

160 

2.42 

20 

135 

280 

2.08 

30 

175 

374 

1.83 

40 

210 

470 

1.70 

50 

255 

580 

1.67 

75 

400 

800 

1.60 

100 

560 

1000 

1.56 

The  commercial  efficiency  of  transformers  at  full  load  ranges 
from  95  to  98  per  cent  or  better  in  machines  of  from  10  kilo- 
watts to  the  large  sizes  of  several  thousands  of  kilovolt- 
amperes capacity.  In  smaller  sizes  the  full  load  efficiency 
becomes  materially  smaller  and  at  one  or  two  kilowatts  capa- 
city it  may  not  exceed  92  or  98  per  cent.  Commercial  trans- 
formers designed  for  the  standard  lighting  frequency  of  60 
cycles  per  second  are  apt  to  have  a slightly  higher  efficiency 
than  those  designed  for  25  cycles  per  second,  and  likewise 
transformers  designed  for  2200  volts  in  the  high  voltage  coil 
(whether  primary  or  secondary)  usually  have  a slightly  higher 
efficiency  than  those  designed  for  higher  voltages.  In  Fig. 

289  the  curves  A and  B represent  the  full  load  efficiencies  of  a 
standard  line  of  500  Kva.  transformers  designed  for  voltages  in 
the  high  voltage  coil  of  2200,  6600,  11,000,  16,500,  22,000, 
33,000,  and  44,000  volts  respectively.  Curve  A is  for  trans- 
formers designed  to  operate  on  60  cycles  per  second  ; and  B 
for  those  operating  on  25  cycles  per  second.  The  curve  in  Fig. 

290  is  the  commercial  efficiency  curve  of  one  of  these  500 
Kva.  transformers  designed  for  2200  volts  in  the  high  voltage 
coil  and  a frequency  of  60  cycles  per  second.  The  curve  is 
typical  of  good  efficiency  curves  for  transformers  of  this  size. 
It  is  noted  in  this  figure  that  the  efficiency  at  ^ load  is  only 
about  2 per  cent  less  than  that  at  full  load,  and  in  this  regard 
it  is  not  far  from  the  performance  of  other  well-designed  trans- 
formers, except  those  of  very  small  capacities,  — five  kilowatts 


MUTUAL  INDUCTION,  TRANSFORMERS 


521 


or  less  capacities,  — which  show  relatively  lower  efficiencies 
at  light  loads. 


100 


99 


98 


96 


95 


94 


__  A 

B 

5000 


10000 


15000  20000  25000  30000 

TRANSFORMER  VOLTAGE 


35000  40000  45000 


Fig.  289.  — Curves  showing  the  Full  Load  Commercial  Efficiencies  of  the  Same  Size 
and  Style  of  Transformers  when  designed  for  Various  Voltages  and  for  Frequencies 
of  60  and  25  Cycles  respectively. 


Fig.  290.  — Curve  of  Commercial  Efficiency  of  a 500-Kva.,  60-cycle,  2200-volt  (High 
.Voltage  Coil ) Transformer. 


522 


ALTERNATING  CURRENTS 


The  weight  efficiency,  that  is,  the  weight  of  transformers 
without  their  containing  cases  or  tanks  per  kilowatt  of  rated 
capacity,  is  from  7 to  20  pounds  for  transformers  of  500  Kva. 
capacity  and  over.  This  increases  by  50  to  100  per  cent 
as  the  size  decreases  to  50  Kva.  or  thereabouts,  and  from  there 
to  10  Kva.  capacity  the  weight  increases  by  10  to  25  per 
cent  further.  For  smaller  sizes,  especially  those  less  than  5 
Kva.,  the  complete  weight  per  kilo-volt-ampere  of  rated  capacity 
including  oil  and  cases,  may  be  as  much  as  eight  or  more  times 
that  of  transformers  of  500  Kva.  capacity.  Consequently, 
large  transformers  are  very  much  less  expensive  to  build  than 
small  ones  per  unit  of  output.  The  weight  of  a transformer  is 
dependent  not  only  on  the  design,  but  also  upon  the  frequency 
and  voltage  of  the  circuit  with  which  it  is  intended  to  be  used ; 
for  low  frequencies  require  a larger  total  magnetic  flux  to  in- 
duce a given  voltage  in  a given  number  of  turns  in  the  wind- 
ings and  hence  they  require  larger  cores  or  greater  windings ; 
very  high  voltages  require  more  insulation  space  to  be  allowed 
in  association  with  the  windings ; and  very  low  voltages 
require  the  use  of  inconveniently  thick  conductors  or  conductors 
composed  of  multiples  of  parallel  elements.  The  ratio  of  weight 
of  copper  to  weight  of  iron  varies  from  about  50  to  80  per  cent, 
when  the  design  follows  the  usual  practice. 

The  regulation  of  transformers  is  of  the  highest  importance, 
especially  where  carbon  filament  incandescent  lamps  are  at- 
tached to  the  secondary  circuit,  as  such  lamps  are  liable  to  serious 
injury  when  subjected  to  varying  voltages.  A closely  constant 
voltage  for  what  is  termed  constant  voltage  apparatus  is  al- 
ways desirable,  though  the  percentage  permissible  variation  is 
dependent  somewhat  upon  the  character  and  duty  of  the  load. 
The  accepted  general  definition  of  “ regulation  ” of  a machine  or 
apparatus  in  regard  to  some  characteristic  quantity  (such  as 
terminal  voltage,  current,  or  speed)  is  “the  ratio  of  the  deviation 
of  that  quantity  from  its  normal  value  at  rated  load  to  the 
normal  rated  load  value,”*  and  for  a constant  voltage  trans- 
former it  is  specifically : “ the  ratio  of  the  rise  of  secondary 
terminal  voltage  from  rated  non-inductive  load  to  no  load  (at 
constant  primary  impressed  terminal  voltage)  to  the  secondary 

*Amer.  Inst,  of  Elect.  Eng.,  Standardization  Rules,  No.  187,  Trans..  1907, 
p.  1809. 


MUTUAL  INDUCTION,  TRANSFORMERS 


523 


terminal  voltage  at  rated  load.”  * The  regulation  is  dependent 
upon  the  drop  of  primary  voltage  due  to  the  leakage  reactance 
and  the  resistance  in  the  primary  and  secondary  coils  of  a 
transformer,  as  may  be  seen  by  a study  of  the  internal  reactions 
of  transformers  given  earlier,  f The  total  rise  of  the  secondary 
terminal  voltage  when  the  load  changes  from  non-reactive  full 
load  to  no  load,  and  the  primary  impressed  voltage  is  maintained 
constant,  is 

ud  = [(u+  i2ny  + /22x2]  * - e, 

where  1^  is  the  full  load  secondary  current,  H is  the  sum  of  the 
primary  and  secondary  coil  resistances,  the  former  reduced  to 
the  secondary  equivalent,  and  X is  the  sum  of  the  leakage  re- 
actances of  the  primary  and  secondary  coils,  the  former  reduced 
to  the  secondary  equivalent,  and  E is  the  secondary  terminal 
voltage  at  rated  full  load.  The  drop  due  to  the  primary  exciting 
current  is  neglected  here.  This  formula  becomes  evident  from 
a study  of  the  loci  of  the  earlier  article  to  which  reference  has 
just  been  made. 

The  regulation  with  a load  of  unity  power  factor  (as  called  for 
in  the  foregoing  definition)  varies  from  less  than  1 per  cent  to 
about  2 per  cent  in  transformers  of  10  Kva.  capacity  and  greater, 
and  may  increase  to  3 or  4 per  cent  in  the  very  small  sizes. 
The  regulation  obtained  when  the  power  factor  of  the  load 
is  other  than  unity  is  impaired  if  the  load  is  inductive, 
but  improves  with  a load  exhibiting  capacity  reactance  until 
the  effect  of  the  leakage  reactance  of  the  coils  is  neutralized 
and  then  with  further  increase  of  capacity  reactance  it  becomes 
poorer.  With  inductive  reactance  in  the  load,  the  regulation 
is  always  poorer  than  when  the  load  is  of  unity  power  factor. 
As  inductive  loads  must  ordinarily  be  reckoned  with  in  com- 
mercial electric  circuits,  it  is  common  for  buyers  of  transformers 
to  specify  the  regulation,  not  only  for  loads  having  unity  power- 
factor,  but  also  for  loads  in  which  the  power  factor  is  .9,  .8, 
and  sometimes  down  to  .6.  If  the  constants  of  a transformer 
are  known,  it  is  easy  to  compute  the  regulation  for  any  power 
factor  of  the  load  by  constructing  transformer  diagrams  or  using 
the  transformer  formulas.  | 


*Ibid.,  No.  197,  p.  1810. 


t Art.  123. 


t Arts.  117-118. 


524 


ALTERNATING  CURRENTS 


138.  Polyphase  Transformers.  — A saving  in  the  amount  of 
material  used,  and  therefore  a reduction  of  the  cost  of  con- 
struction and  an  improvement  of  the  operating  efficiency,  may 
be  effected  for  polyphase  transformation  by  combining  the 
magnetic  circuits  of  the  individual  transformers  in  the  several 
phases  in  a manner  which  is  analogous  to  that  in  which  poly- 
phase electric  circuits  are  combined  into  common  circuits. 

Figure  291  represents  a quarter-phase  transformer  with  a 
combined  magnetic  circuit.  Since  the  phases  of  the  magnetic 
fluxes  in  the  two  halves  of  the  transformer  are  90°  apart  (the 
phases  of  the  impressed  voltages  being  90°  apart  and  here 

assumed  sinusoidal),  the  re- 
sultant magnetism  in  the 
middle  tongue  is  their  vec- 
tor sum  and  is  V2  times  as 
great  as  that  in  either  of 
the  cores  under  the  wind- 
ings. Therefore  this  cen- 
tral tongue  should  have  V2 
times  as  great  a cross  sec- 
tion as  the  remainder  of  the 
magnetic  circuit.  There  is 
a saving  of  iron  in  the  com- 
bined transformer,  as  com- 
pared with  two  independent 
transformers,  which  is  equal 
to  ( V2  — 1)  times  the  weight 
of  the  central  tongue.  This  saving  is  further  emphasized  by  the 
advantage  of  the  small  space  occupied  and  the  convenient  form 
obtained  by  inclosing  this  equivalent  of  two  transformer  units 
in  one  case.  In  this  illustration,  the  high  voltage  external  cir- 
cuits are  shown  with  a common  return  wire,  and  the  low  vol- 
tage external  circuits  are  shown  with  independent  wires  ; but  it 
is  obvious  that  the  use  of  independent  circuits  requiring  four 
wires  for  the  two  phases,  or  the  use  of  three  wire  circuits 
utilizing  one  wire  for  the  common  return,  are  free  alternatives 
for  use  with  the  primary  and  secondary  circuits. 

A similar  combination  may  be  effected  in  tri-phase  trans- 
formers, and  the  magnetic  circuits  may  be  coupled  in  either  the 
wye  or  the  delta  arrangement.  Figure  292  shows  a conventional 


Fig.  291.  — Diagram  of  Circuits  and  Core  of  a 
Quarter-phase  Transformer. 


MUTUAL  INDUCTION,  TRANSFORMERS 


525 


diagram  of  a tri-phase  transformer  of  a form  early  made  by  several 
manufacturers,  in  which  the  magnetism  in  the  yokes  DD',  which 
join  the  cores  A,  B , and  (7,  is  1/V3  times  as  great  as  that  in  the 
cores.  These  yokes  make  a delta  coupling  for  the  magnetic 


Fig.  292.  — Diagram  of  Circuits  and  Core  of  a Tri-phase  Transformer  with  a Yokes. 

circuits.  Figure  293  shows  a similar  diagram  of  a tri-phase 
transformer,  in  which  the  yokes  are  joined  so  as  to  make  a wye 
coupling  of  the  magnetic  circuits.  These  constructions  allow 
a considerable  economy  in  use  of  materials  in  comparison  with 


Fig.  293.  — Diagram  of  Circuits  and  Core  of  a Tri-phase  Transformer  with  Y Yokes. 

separate  transformers  in  the  three  phases,  on  account  of  the 
fact  that  the  magnetic  circuit  through  the  core  under  each 
winding  is  completed  through  the  cores  under  the  other  two 
windings. 

In  Fig.  294  is  shown  a diagram  of  circuits  and  core  for  a 
common  type  of  tri-pliase  transformer.  To  obtain  an  expres- 
sion for  the  instantaneous  magnetic  fluxes  in  the  three  cores 


526 


ALTERNATING  CURRENTS 


W,  X,  and  y,  shown  in  Fig.  294,  assume  the  magnetic  reluc- 
tance in  the  path  from  A to  A'  indicated  by  the  U-shaped  dotted 
line  through  the  core  W and  the  path  indicated  by  the  corre- 
sponding U-shaped  dotted  line  through  the  core  Y to  be  equal, 
and  of  constant  value  Pw ; also  let  the  reluctance  of  the  path 
directly  from  A to  A'  through  X he  represented  by  P Con- 
sider that  sinusoidal  alternating  exciting  currents,  120°  apart 


Fig.  294.  — Diagram  of  Circuits  and  Core  of  a Tri-phase  Transformer  having 
Straight  Yokes. 

in  their  phases,  flow  in  the  three  exciting  windings,  and  let  <f>w, 
(f)x,  and  (j)y  represent  values  of  the  magnetic  fluxes  in  the  cores 
W,  X , and  Y,  respectively,  at  a given  instant,  and  mw,  mx,  and 
my  represent  the  corresponding  instantaneous  values  of  the 
magneto-motive  forces  exerted  in  the  aforenamed  paths  through 
the  three  cores.  Then,  using  the  same  conventions  in  regard  to 
the  algebraic  signs  of  the  instantaneous  values  of  magneto- 
motive forces  and  fluxes  in  the  several  parts  of  the  transformer 
as  are  set  forth  in  Art.  103  for  three-phase  electric  circuits, 
the  following  simultaneous  equations  represent  the  conditions  : 

mw  — mx  = cf)lcP,c  - <j>xPx, 
mw  — my  = 4>irPlc  — 
mx-my  = 4>UPX  - <$>yPw, 

flA  + 4>x  + = 


MUTUAL  INDUCTION,  TRANSFORMERS 


527 


Solving  these  equations  gives  the  values  of  cf>w,  (px, 
terms  of  mw,  mx , my , Pw,  and  Px,  as  follows  : 

and  <f)y  in 

, _ mw  — mx  Px(rnw  - my) 

P +2P  P (P  + 2P) 

(1) 

, _ mw  — my  2 (mw  — »q) 

P + 2P  P +2  P 

(2) 

, mv-mx  Px(mu-mw ) 

Pw  + 2 Px  PJPW  + 2 Px) 

(3) 

Putting  mw  = rt/sin  a, 

mx  = al sin  («  — 1 20°), 
my  - al  sin  («  — 240°), 

a being  a constant,  these  equations  become, 

</>»  = p a fsin  a - sin  («  - 120°) 

w + 1 + PJPw\s\n  a — sin  (a  — 240°)] } (4) 

<f>x=  pa\p  <2  sin  («  — 120°)  — sin  («  — 240°)  — sin  (5) 

4>v  = Jsin(a-  240°)-  sin(a  - 120°) 

w^r~‘  x + Pz/Pu,[sin(a  — 240°)  — sin  a]|.  (6) 


The  hysteretic  angle  of  lead  being  zero  under  the  assumption 
made,  these  fluxes  come  to  their  respective  maxima  when 
a = 90°,  a=  210°,  and  a=  330°,  and  have  the  numerical  values, 

= O + A/A)  (7) 

w "4  x 


cp 

1 X 


Sal 

P.  + 2P,' 


(8) 


These  values  are  obtained  on  the  assumption  that  sinusoidal 
magneto-motive  forces  of  equal  amplitudes  and  differing  suc- 
cessively in  phase  by  120°  are  impressed  by  the  coils  in  the 
three  branches  of  the  magnetic  circuit,  and  also  on  the  assump- 
tion that  Px  differs  from  Pw  and  Py,  but  that  the  last  two  are 
equal  to  each  other. 

It  is  obvious  from  these  formulas,  that  with  equal  magneto- 
motive forces,  the  value  of  dq  is  larger  than  the  value  of  dq 
and  dq,  in  case  Px  is  smaller  than  Pw  on  account  of  the  shorter 


528 


ALTERNATING  CURRENTS 


length  of  the  middle  magnetic  circuit.  The  maximum  fluxes 
<£>„,,  d>„  are  all  equal  to  one  another  when  Px  = Pw  = Py ; 

but  if  Px  is  negligible  compared  with  Pw , <t>z  becomes  twice  as 
large  as  and  dq,. 

Under  the  usual  conditions  of  transformer  operation,  equal 
voltages,  120°  apart  in  their  phases,  would  be  impressed  on  the 
three  equal  primary  windings.  Under  these  circumstances, 
equal  fluxes  would  be  set  up  in  the  three  branches  of  the 
magnetic  circuit,  and  equations  (7)  and  (8),  which  give  the 
relations  between  the  fluxes  and  exciting  currents,  show  that 
the  exciting  currents  must  differ  under  these  circumstances 
unless  Px  = Pw  = Pv.  Such  a difference  of  currents  causes  an 
unbalancing  of  the  supply  circuits.  Also  on  account  of  the 
difference  of  form  of  the  magnetic  leakage  path  around  the 
middle  limb  compared  with  the  leakage  paths  around  the  two 
outer  limbs,  unbalancing  is  felt  in  the  secondary  voltage. 

To  partially  avoid  this  unbalancing,  the  middle  leg  may  be 
made  of  smaller  cross  section  than  the  other  two  legs,  so  that 
Px  is  made  more  nearly  equal  to  Pw  and  Py , but  this  is  likely 
to  increase  the  core  losses  on  account  of  the  increased  magnetic 
density  in  that  part  of  the  core,  and  it  also  changes  the  power 
factor  of  the  exciting  current  of  that  leg  compared  with  the 
currents  exciting  the  other  legs.  As  the  yokes  which  complete 
the  magnetic  circuits  at  the  ends  of  the  limbs  obviously  carry 
the  same  maximum  flux  as  the  limbs  themselves,  if  magnetic 
leakage  is  neglected,  the  yokes  are  usually  made  of  the  same 
cross  sections  as  the  outer  limbs.  The  same  advantages  in 
respect  to  reduced  weight  and  bulk  and  reduced  exciting 
current  relate  to  this  form  of  construction  as  to  the  construc- 
tion with  wye  and  delta  magnetic  circuits. 

The  magneto-motive  forces  of  the  three  primary  coils  being 
out  of  phase  with  each  other,  magneto-motive  force  pulsations 
between  the  points  8 and  T,  and  U and  V are  created  which 
tend  to  set  up  leakage  fluxes  between  these  points,  which  is  in 
addition  to  the  ordinary  magnetic  leakage  between  the  primary 
and  secondary  coils.  Pulsations  of  the  third  harmonic  are  most 
prominent  and  are  in  the  same  phase  and  the  same  direction 
through  their  respective  cores.*  Quite  large  additional  third 
harmonic  currents  may  therefore  flow  around  the  mesh  made  by 

* Art.  198. 


MUTUAL  INDUCTION,  TRANSFORMERS 


529 


the  primary  coils  if  they  are  delta  connected  as  shown  in  Fig.  294. 
If,  however,  the  coils  are  connected  in  wye  without  a neutral 
return  wire,  these  currents  comprising  the  third  harmonic  can- 
not flow  and  the  resulting  leakage  magnetism  is  absent.  If 
the  neutral  point  of  the  wye  is  connected  to  a neutral  or  fourth 
wire  of  the  supply  system,  the  currents  comprising  the  third 
harmonic  will  flow  through  the  three  lead  wires  and  coils  in 
parallel  relation  and  complete  their  circuit  through  the  neutral 
wire. 

When  the  three-phase  transformer  is  of  the  shell  type  (in 
which  the  windings  are  embedded  in  the  iron),  as  in  Fig. 
295,  and  the  primary  windings  are  all  connected  in  the  same 
relative  direction,  the  result- 
ant magneto-motive  force  act- 
ing along  the  whole  length  of 
the  common  core  is 

Ma  + Mb  + Mc  = 0, 

where  MA,  MB,  and  Mc  are 
the  magneto-motive  forces  in 
the  coils  A , B , and  (7,  in  case 
the  magnetizing  currents  are 
assumed  to  be  sinusoidal,  equal 
in  scalar  value,  and  120°  apart 
in  phase.  This  means  that 
the  vector  sum  of  the  mag- 
neto-motive forces,  as  in  the 

previous  case,  equals  zero,  but  Fig-  295. — Diagram  of  In-phase  Shell 
1 l . 1 . Type  Transformer. 

in  this  case  any  two  are  m 

series  instead  of  parallel  with  reference  to  the  third  ; also, 
®A  + ®B+  $c  = °, 

where  d>A,  d>5,  and  d>c  respectively  represent  corresponding 
vector  values  of  the  three  magnetic  fluxes. 

The  distribution  of  the  magnetism  in  the  core  is  as  follows : 
The  flux  in  Y is  the  combination  of  the  fluxes  threading  coils 
C and  B , and  that  in  X the  combination  of  the  fluxes  threading 
coils  B and  A.  As  the  vector  fluxes  threading  coils  C and  B 
are  numerically  equal,  but  differ  in  phase  by  120°,  the  flux  in  Y 


530 


ALTERNATING  CURRENTS 


is  (neglecting  magnetic  leakage)  equal  to  vector  (7-flux  plus 
vector  ,5-flux  reversed.  That  is.  the  vector  addition  is  made 
in  the  same  manner  as  for  three-phase  electric  circuits.*  There- 
fore Y,  and  for  the  same  reason  X,  must  carry  a flux  equal 
to  V3  times  the  flux  threading  either  of  the  coils.  The  maxi- 
mum flux  in  the  shell  at  W,  Z,  and  H has  the  same  value 
(neglecting  magnetic  leakage)  as  the  maximum  flux  threading 
the  coils. 

When  the  middle  coil  B in  Fig.  295  is  reversed  relatively  in 
respect  to  A and  (7,  the  magneto-motive  force  formula  becomes 

Mr  = MA-MB  + Mc. 

The  vectors  on  the  right-hand  side  of  this  formula  are  now 
only  60°  apart  in  phase  and  the  scalar  value  of  their  resultant 
Mr  — 2 M,  where  M is  the  scalar  value  of  the  vector  magneto- 
motive force  in  each  coil.  But  the  fluxes  in  all  of  the  cores 
must  be  equal  in  scalar  value  and  60°  apart  in  phase  in  order 
to  produce  the  proper  counter-voltages  in  the  primary  coils. 
Therefore  the  flux  due  to  JEfr,  or 

A>r  = 

where  <3?^,  <3>5,  and  <3>c  are  the  fluxes  threading  the  coils  A,  B. 
and  C.  In  order  to  maintain  the  proper  counter-voltages  in  the 
windings  and  vector  sum  of  fluxes  in  the  main  core,  flux  d>A  is 
set  up  in  the  magnetic  circuit  AWITXA,  <3>fi  in  BXHYB . 
and  <3>c  in  CYHZC.  Fluxes  <£>4  and  <t> B are  60°  apart  in  phase 
in  the  central  core  or  tongue,  so  that  they  are  120°  apart  in 
phase  in  X.  The  tensor  of  their  sum  is  then  equal  to  the  scalar 
value  of  either  of  the  two.  The  same  conditions  exist  in  Y,  so 
that  the  combined  cross  section  of  iron  at  A,  B,  (7,  W.  X . Y,  Z. 
and  H may  be  made  uniform  for  the  same  induction  in  all  parts 
of  the  iron.  There  is  thus  some  saving  obtained  in  material,  or 
reduction  in  the  reluctance  of  the  core,  by  reversing  the  mid- 
dle coil  connections.  The  exciting  current  is  the  same  in  each 
phase  as  would  be  required  in  three  single-phase  transformers 
having  equal  core  reluctances. 

When  the  voltages  or  currents  or  both  are  unbalanced,  the 
conditions  still  hold  that  the  magnetic  flux  for  each  phase  must 


* Art.  103. 


MUTUAL  INDUCTION,  TRANSFORMERS 


531 


be  such  that  it  will  produce  the  proper  counter-voltage,  and 
hence  the  flux  under  each  coil  remains  proportional  to  the 
counter-voltage  in  that  phase ; and  the  three  fluxes,  however 
different  in  value,  will  combine  vectorially  according  to  the 
same  laws  as  where  the  circuits  are  balanced. 

139.  Some  Constructive  Features  of  Constant  Voltage  Trans- 
formers. — The  formula 

™ _ V2  Trn<t>f 
1 108 

contains  the  four  quantities  Uv  n , <E>,  and  /,  which  may  be 
varied,  but  in  practice  and  f are  usually  fixed  by  the  re- 
quirements of  the  service,  and  only  n and  <1>  may  be  varied. 
Their  relative  variation  is  also  limited  by  the  fact  that  the 
product  n<&  is  fixed  when  JEX  and  f are  fixed.  In  designing  a 
transformer,  the  relative  values  of  n and  <3>  depend  upon  the 
relative  weights  which  it  is  considered  desirable  to  assign  to  the 
copper  and  iron  in  the  transformer.  When  the  magnetic  density 
which  is  safe  to  use  in  the  core  has  been  chosen,  the  cross  section 
of  the  core  depends  directly  upon  fl?  ; while  n,  and  therefore 
the  weight  of  copper,  assuming  that  the  size  of  wire  to  be  used 
has  been  decided  upon,  is  inversely  dependent  upon  the  same 
quantity.  The  weight  of  iron  depends  upon  the  cross  section 
and  length  of  the  magnetic  circuit,  and  must  be  limited  so 
that  the  losses  occurring  in  the  iron  of  available  quality  shall 
not  reduce  either  the  full  load  efficiency  or  the  all-day  efficiency 
below  a reasonable  figure. 

The  number  and  length  of  the  turns  ( n ) of  wire  used  in 
each  winding  and  the  size  of  the  conductors  which  are  chosen 
for  the  purpose  of  preventing  injurious  heating,  objectionable 
reduction  of  full  load  efficiency,  or  poor  regulation,  jointly 
determine  the  required  weight  of  copper.  The  density  of 
current  in  the  windings  varies  widely,  from  400  to  over  2000 
circular  mils  cross  section  per  ampere.  This  density  is  largely 
dependent  upon  the  form  of  the  coils.  These  are  seldom 
made  so  that  the  heat  liberated  by  the  12R  loss  must  travel 
more  than  one  half  an  inch  before  reaching  the  radiating 
surface  of  the  coil.  It  is  not  unusual  to  make  the  density 
somewhat  smaller  in  the  low  voltage  winding  than  in  the 
high  voltage  winding,  and  the  values  in  the  best  transform- 


532 


ALTERNATING  CURRENTS 


ers  frequently  fall  between  1000  and  1500  circular  mils  per 
ampere  for  the  high  voltage  winding  and  between  1200  and 
2000  for  the  low  voltage  winding.  On  the  other  hand,  some 
designers  make  the  density  of  current  greater  in  the  low  vol- 
tage windings,  while  others  make  the  density  about  the  same  in 
both.  As  the  high  voltage  conductors  carry  less  current  and 
are  therefore  of  smaller  cross  section  than  the  low  voltage  con- 
ductors, the  insulation  of  the  high  voltage  winding  takes  up 
more  space  in  proportion  to  the  space  occupied  by  the  con- 
ductors ; and  this  condition  becomes  emphasized  when  the  vol- 
tage exceeds  a few  thousands  of  volts,  on  account  of  the  exag- 
gerated thickness  of  insulation  required  for  the  high  voltage. 
This  heavier  insulation  also  makes  it  more  difficult  for  the  heat 
generated  to  escape,  and  it  is  often  necessary  to  divide  the  high 
voltage  winding  into  a number  of  thin  coils  separated  by  ducts 
for  the  circulation  of  air  or  oil. 

The  I2R  loss  in  transformers  at  full  load  is  ordinarily  equal 
to  from  per  cent  to  3-|-  per  cent  of  the  full  load  capacities. 
This  is  divided  with  approximate  equality  between  the  primary 
and  secondary  windings.  Sometimes  this  loss  is  permitted  to 
reach  5 per  cent,  but  in  the  better  transformers  it  is  more 
often  between  1 per  cent  and  3 per  cent.  The  percentage  of 
loss  which  is  allowable  depends  primarily  upon  the  trans- 
former efficiency  and  regulation  which  are  desired,  and  secon- 
darily upon  the  duty  to  be  performed,  the  voltage,  and  the 
frequency. 

Transformers  are  of  two  general  classifications,  termed  the 
Core  type  and  the  Shell  type,  these  designations  depending  upon 
the  manner  in  which  the  windings  and  the  iron  core  are  disposed 
with  respect  to  each  other.  Both  forms  seem  to  lend  them- 
selves satisfactorily  to  the  design  of  transformers  for  ordinary 
duty,  and,  in  fact,  there  is  no  definite  line  of  demarcation  be- 
tween them.  In  general  terms,  core  type  transformers  may 
be  defined  as  those  in  which  each  of  the  principal  branches  of 
the  magnetic  circuit  is  embraced  by  windings,  as  illustrated  in 
Figs.  47  and  296,  while  shell  type  transformers  are  those  in 
which  the  return  branches  of  the  magnetic  circuit  embrace  the 
windings,  as  illustrated  in  Figs.  300  and  301. 

Laminations  for  the  cores  are  made  in  various  shapes.  Fig- 
ure 296  represents  the  core  and  coils  of  a core  type  transformer, 


MUTUAL  INDUCTION,  TRANSFORMERS 


533 


and  Fig.  297  shows  a pair  of  the  lami- 
nations or  stampings  of  which  the  core 
is  made  up.  When  the  stampings  are 
of  this  form,  they  are  built  up  into  a 
core  within  the  iinished  coils.  In  every 
other  layer  of  the  core  the  stampings 
are  reversed  end  for  end  so  that  they 
break  joints.  This  is  for  the  purpose 
of  reducing  the  magnetic  reluctance 
of  the  magnetic  circuit  to  a minimum. 

It  is  important  that  joints  in  the  mag- 
netic circuit  shall  be  as  few  as  possible, 
and  be  so  made  as  to  be  of  low  reluc- 
tance, as  otherwise  the  exciting  current 
may  be  caused  to  be  of  undesirable 
magnitude.  Rolled  iron  or  steel  sheets 
as  they  come  from  the  rolling  mills  are  usually  covered  by  a 
tough  and  closely  adherent  coating  or 
thin  scale  of  magnetic  oxide  of  iron, 
which  is  developed  in  the  process  of 
manufacture  of  the  sheets.  This  is  a 
poor  electrical  conductor,  and  since  it 
sticks  to  the  stamping,  it  usually  affords 
sufficient  insulation  between  the  lamina- 
tions to  prevent  eddy  currents  from  becom- 
ing excessive.  However,  to  give  double 
assurance,  it  is  usual  to  dip  the  stampings 
into  a liquid  varnish  or  enamel  and  then 
bake  them  to  solidify  the  coat  before  build- 
ing them  into  a core. 


Fig.  296.  — Core  Type  Trans- 
former, Core  and  Coils. 


Fig.  297.  — Laminations  In  Fig.  298  is  shown  a leg  of  the  core 

for  Transformer  shown  a iarge  transformer  which  is  composed 

of  rectangular  laminations  built  into  a 
cross  section  having  a cruciform  shape.  This  is  for  a core 
type  transformer  such  as  is  shown 
in  Fig.  47.  The  complete  core  com- 
prises two  such  legs,  each  covered 
with  windings,  and  connecting  yokes 
completing  the  magnetic  circuit  at  former  Core  buiIt  up  of  Rect. 
their  ends.  The  crossbars  con-  angular  Laminations. 


534 


ALTERNATING  CURRENTS 


necting  the  ends  of  the  legs  are  also  made  of  rectangular 
laminations  which  alternately  slip  between  and  butt  against 
those  on  the  leg  shown,  every  other  one  on  the  leg  being  made 
short  to  give  space  for  this  purpose,  or  the  dovetailing  may  be 

done  by  bunches 
of  laminations 
instead  of  by  al- 
ternate sheets. 
The  type  of  coil 
used  for  the  wind- 
ings of  a trans- 


Fig.  299.  — Division  of  a Coil  for  the  Transformer  of  which 
Part  of  the  Core  is  shown  in  Fig.  298. 


former  using  the 


cruciform  core  is  shown  in  Fig.  299.  It  is  composed  of 
a cotton-covered  rectangular  cojiper  strip  wound  edgewise. 
Where  possible  to  insulate  properly  proportioned  rectangular 
wire,  its  use  is  often  desirable,  as  it  occupies  less  space  for 
a given  effective  area  than  round  wire.  It  is  also  frequently 
desirable  to  wind  it  edgewise,  that  is,  with  its  greatest  width 
in  the  plane  at  right  angles  to  the  core,  as  this  construction 
often  makes  it  practicable  to  make  the  complete  coil  with 
only  one  layer  of  the  conductor,  thus  affording  opportunity 
for  efficient  cooling,  and  the  con- 
struction may  be  made  rigid  and 
strong.  Bare  strips  are  sometimes 
wound  in  the  edgewise  coils,  with 
thin  strips  of  vulcanized  fiber,  var- 
nished paper,  or  varnished  cambric 
laid  on  edge  between  the  turns  of 
conductor. 

Figure  300  shows  a shell  type  trans- 
former with  the  laminations  horizon- 
tal. The  wire  in  this  construction  is 
wound  with  its  flat  faces  parallel  to 
the  axis  of  the  coil,  and  the  windings 
are  made  up  of  a number  of  flat  sec- 
tions, each  having  a thickness  equal 
to  the  breadth  of  one  conductor  strip. 

It  is  usual  in  the  larger  trans- 
formers,— 50  Kva.  capacity  and  over,  — to  split  up  the  primary 
winding  into  sections  or  divisions  and  sandwich  between  them 


MUTUAL  INDUCTION,  TRANSFORMERS 


535 


the  secondary  winding,  which  is  also  wound  in  sections.  This 
reduces  magnetic  leakage  and,  when  free  space  is  left  between 
the  coil  sections,  facilitates  the  dissipation  of  heat  by  providing 
opportunity  for  the  circulation  of  air  or  oil. 

Figure  301  shows  a shell  type  transformer  in  which  the 
laminations  and  coils  are  vertical.  In  this  case  the  coils  are 
made  elongated  as  in  Fig.  299,  and  therefore  the  sections  of  the 
primary  and  secondary  windings  are  sandwiched  within  one 
another,  while  in  Fig.  300  the  sections  are  flat  and  wide  and 
lie  side  by  side. 


Fig.  301.  — Shell  Type  Transformer  with  the  Coils  and  Laminations  Vertical. 

From  the  illustrations  it  may  be  observed  that  core  type 
transformers  usually  consist  of  a simple  rectangular  magnetic 
circuit  with  half  of  the  primary  and  secondary  turns  wound 
upon  each  of  two  legs,  and  the  cross  sections  of  the  two  legs 
and  the  two  end  pieces  are  equal.  Likewise  it  may  be  observed 
that  the  core  of  a shell  type  transformer  usually  comprises  a 
central  branch  or  leg  upon  which  the  windings  are  located  and 


536 


ALTERNATING  CURRENTS 


which  is  of  double  the  cross  section  of  the  remainder  of  the 
magnetic  circuit  if  the  latter  is  closed  by  two  branches  as  illus- 
trated in  Figs.  300  and  301. 

Figure  301  a shows  the  construction  looking  down  on  the  core 

and  windings  from  the  top, 


Mica 
Shields y 


Secondary 

Winding 


OH  Duel 


'Oil  Channels 


Fig.  301  a.  — Transformer 
Type. 


of  Intermediate 


of  a transformer  belonging 
to  a type  now  much  used,  in 
which  the  windings  surround 
a central  branch  of  the  mag- 
netic circuit,  and  the  mag- 
netic circuit  is  closed  by 
several  limbs  which  join 
radiating  end  pieces.  This 
arrangement  lias  proved  ad- 
vantageous for  transformers 
of  small  and  medium  capac- 
ities, since  it  affords  windings 
in  which  the  mean  length  of 
turn  is  reasonably  short  and 
at  the  same  time  affords  a magnetic  circuit  of  the  requisite 
cross  section  while  permitting  the  use 
of  a moderate  bulk  and  weight  of  iron. 

This  results  in  satisfactory  regulation 
associated  with  reasonably  good  all-day 
efficiencies.  The  construction  also 
has  some  advantages  in  reducing  the 
total  bulk  of  small  transformers  with- 
out interfering  objectionably  with  the 
opportunities  for  getting  rid  of  the 
heat  produced  by  the  operation.  It 
presents  a little  more  difficulty  in  the 
mechanical  construction  of  the  core, 
more  particularly  in  respect  to  obtain- 
ing a magnetic  circuit  of  the  lowest 
practicable  reluctance  without  undue 
expense. 

Figure  302  shows  a tri-phase  trans- 
former of  the  shell  type  with  three  cores  fig.  302.  — Tri-phase  Traus- 

meeting  at  the  center  for  carrying  the  former  Wlth  the  Ma-'ienc 
^ J ° Circuits  connected  in  V ye 

coils  of  the  three  phases.  It  is  com-  Fashion. 


MUTUAL  INDUCTION,  TRANSFORMERS 


537 


mon  in  this  type  of  transformer  to  have  the  coils  of  the  three 
phases  distributed  on  a single  core,  as  in  the  diagram  of  Fig. 
295.  A core  type  tri-phase  transformer  may  be  arranged  as 
in  the  diagram  of  Fig.  294,  and  the  reluctances  of  the  three 
magnetic  paths  can  be  made  equal  by  arranging  the  limbs  at  the 
corners  of  a triangle,  with  their  magnetic  circuits  joined  in 
either  delta  or  wye,  as  indicated  in  Figs.  292  and  293. 

The  quarter-phase  transformer  core  may  be  built  conven- 
iently as  shown  in  the  diagram  of  Fig.  291,  or  it  may  be  built 
of  the  shell  type,  similiar  to  that  shown  in  Fig.  295. 

The  various  arrangements  in  which  the  iron  of  a transformer 
may  be  disposed,  which  will  give  approximately  the  same  results 
in  operation,  are  quite  numerous,  and  the  particular  one  to  be 
selected  in  any  instance  is  apt  to  turn  upon  the  question  of  the 
cost  of  manufacture  rather  than  the  needs  of  service.  The  first 
and  most  important  element  to  be  dealt  with  in  the  construc- 
tion of  a transformer,  as  in  the  case  of  most  other  electrical 
machinery,  is  the  disposition  of  the  heat  occasioned  by  the 
iron  and  copper  losses.  Upon  the  solution  of  this  problem 
depends  the  rise  of  temperature  of  the  transformer  per  kilovolt- 
ampere of  load  when  in  operation,  and  its  ultimate  safe  capac- 
ity. The  windings  of  transformers  are  usually  embedded 
more  or  less  in  the  iron  cores,  as  is  illustrated  in  the  preceding 
figures  ; and  the  whole  transformer  is  inclosed  in  a waterproof 
iron  case,  as  illustrated  in  Figs.  47,  303,  and  304.  The 
rise  of  temperature  when  in  operation  is  due  to  the  heating 
of  the  structure  by  the  core  losses  and  the  copper  losses.  If 
transformers  were  not  encased,  but  were  placed  naked  in  the 
open  air,  the  entire  external  surface  could  be  assumed  to  be 
effective  in  dissipating  heat  by  radiation  and  convection  ; but 
on  account  of  the  inclosing  case,  convection  of  heat  to  the  ex- 
ternal atmosphere  cannot  take  place  directly,  and  all  the  heat 
must  be  radiated  to  the  wall  of  the  case  or  carried  thereto 
through  the  poor  heat  conductors  which  are  used  to  electrically 
insulate  the  transformer  from  its  case.  The  conditions  there- 
fore point  to  the  conclusion  that,  without  some  special  method 
of  cooling,  for  a given  liberation  of  heat  per  square  centimeter 
of  surface,  the  working  temperature  is  likely  to  be  higher  in 
transformers  than  in  dynamo  field  windings.  In  fact,  to  pre- 
vent excessive  working  temperatures  and  permit  the  highest 


538 


ALTERNATING  CURRENTS 


safe  output  per  pound  of  iron  and  copper,  special  means  of  cool- 
ing are  usually  provided  for  transformers.  These  are  described 
later. 

Transformer  coils  are  usually  designed  to  be  lathe  wound, 
and  these  may  be  very  effectually  insulated  by  a liberal  use  of 
mica,  varnished  cotton  cloth,  fiber,  and  wood.  It  is  therefore 
possible  to  safely  run  transformers  with  the  windings  at  a con- 
siderably higher  temperature  than  dynamos.  Sixty  degrees 
centigrade  (108°  F.)  may  be  set  as  a maximum  limit  to  the 
safe  rise  in  temperature  caused  by  operation,  though  so  high  a 
temperature  is  undesirable,  both  because  of  depreciation  of  the 
insulating  materials  and  the  tendency  of  the  core  to  age  unless 
a non-aging  steel  alloy  is  used.  A high  temperature  limit  has 
also  a marked  disadvantage  in  causing  an  undue  impairment 
of  the  regulation  when  the  transformer  is  hot,  which  results 
from  the  increased  resistance  of  the  windings.  Many  trans- 
formers still  in  operation  exceed  the  temperature  limit  named, 
but  most  of  the  best  types  do  not  exceed  40  ° centigrade  rise. 
As  the  rise  in  temperature  also  increases  the  electrical  resist- 
ance of  the  iron  core,  it  decreases  the  eddy  current  loss,  so  that, 
as  suggested  by  Elihu  Thomson,  it  was  early  considered 
advantageous  to  have  the  core  of  a transformer  operated  at  a 
high  temperature  while  the  windings  were  kept  cool.  This 
could  not  be  conveniently  arranged  in  small  transformers,  but 
the  cooling  of  the  conductors  of  very  large  transformers  has 
been  experimentally  effected  by  making  the  conductors  tubular 
and  passing  a cool  liquid  through  them. 

It  is  practically  impossible  to  fix  any  averages  for  the  external 
surface  of  transformers  needed  per  watt  lost  in  the  core  and  wind- 
ings, on  account  of  the  very  varied  arrangements  of  the  coils  with 
reference  to  the  core,  and  the  effect  of  the  containing  case.  In 
good  commercial  designs  of  transformers  the  superficial  area  of 
core  and  windings  varies  from  3 to  7 square  inches  per  watt  of 
energy  to  be  dissipated.  For  transformers  of  small  and  medium 
capacities  it  is  usual  to  make  the  design  as  compact  as  possible, 
and  no  particular  trouble  from  heating  is  experienced  if  the 
losses  are  not  excessive  when  judged  from  the  criteria  of 
efficiency  and  regulation,  provided  the  inclosing  case  is  filled 
with  oil  for  the  purpose  of  improving  the  facility  with  which 
the  heat  can  escape  to  the  outer  surface  of  the  case.  A highly 


MUTUAL  INDUCTION,  TRANSFORMERS 


539 


refined  petroleum  oil  carefully  filtered  and  dried  is  used  for  this 
purpose,  and  is  commonly  called  Transformer  oil,  on  account  of 
the  principal  object  of  its  use.  Such  oil  also  posesses  high  insu- 
lating qualities,  which  is  an  advantage  in  the  transformers. 
Transformers  with  which  oil  is  associated  for  cooling  purposes 
are  called  Oil-cooled  transformers.  Those  without  oil  are  some- 
times called  Dry-core  transformers.  As  transformers  increase 
in  size,  the  weight  per  unit  of  capacity  decreases,  and  the 
superficial  area  per  unit  available  for  dissipating  heat  decreases 
at  a still  more  rapid  rate,  since  the  bulk  and  weight  are  pro- 
portional to  the  cube  of  the  linear  dimensions,  and  the  super- 
ficial area  is  proportional  to  only  the  square  of  the  linear 
dimensions ; and  some  device,  such  as  circulating  the  oil  within 
the  case  and  through  ducts  in  the  core  and  between  the  divisions 
of  the  windings,  or  blowing  air  through  ducts  in  the  core,  must 
be  provided  for  forcing  the  more  rapid  conveyance  of  heat  from 
the  core  and  windings.  Otherwise  the  heating  would  be  ex- 
cessive unless  the  transformer  is  of  unreasonably  great  and  ex- 
pensive bulk.  Dry  core  transformers,  arranged  for  cooling  by 
a blast  of  air,  are  called  Air-cooled  transformers. 

A small  transformer  will  evidently  cool  more  readily  than 
a large  one  of  the  same  type  and  design.  Thus,  in  a number  of 
transformers  of  different  capacities  having  the  same  efficiencies, 
losses  per  pound  of  copper  and  iron,  and  voltages,  and  of  similar 
design,  if  we  let  A stand  for  linear  dimensions,  A for  superficial 
area,  W for  weight,  KL  for  full  load  losses,  and  Kva  for  rated 
full  load  output,  we  have 

TFxA*;  ioci2; 

Kl  x W x A3  x A 2,  approximately ; 

4.  4 

Kva  oc  A4  oc  IF3  x x A2,  approximately. 

It  is  evident  that  Kva  varies  approximately  as  A4,  since  when 
the  dimensions  are  varied  the  cross  section  of  the  magnetic  cir- 
cuit and  hence  the  magnetic  flux  is  varied  as  A2.  With  fixed 
voltage,  the  number  of  turns  in  the  windings  therefore  changes 

proportionately  to  ^ and  the  space  in  which  the  windings  go 

changes  proportionately  to  A2.  Hence  with  winding  space 


540 


ALTERNATING  CURRENTS 


changed  in  proportion  to  L 2 and  number  of  turns  in  pro- 


portion to  — , the  area  of  each  conductor  may  be  changed  in 


proportion  to  X4,  and  its  current-carrying  capacity  correspond- 
ingly changed,  provided  the  necessary  insulating  space  can  be 
reserved  and  the  heat  produced  by  the  PR  losses  at  full  load 
can  be  carried  off. 

It  is  seen,  therefore,  that  transformers  of  large  sizes  should 
have  normally  less  weight  per  kilowatt  capacity,  full  load  losses 
lower  in  proportion,  and  higher  efficiency  than  those  of  smaller 
size,  — which  deductions  are,  in  fact,  borne  out  in  practice. 
But  the  dissipation  of  heat  in  the  larger  transformers  is  more 
difficult,  since  KL  oc  X3  oc  AK  and  therefore  the  area  exposed  for 
cooling  in  similar  transformers  of  different  sizes  varies  only 
with  the  f-  power  of  the  losses,  when  the  losses  are  taken  to  be 
proportional  to  the  weights  of  the  transformers,  and  it  also 
varies  as  the  square  root  of  the  rated  transformer  capacity 
given  in  kilovolt-amperes.  Hence  it  is  evident  that  if  a small 
transformer  of  a given  form  heats  at  full  load  to  the  maximum 
temperature  allowed,  say  40°  centigrade,  a transformer  of  ten 
times  the  size  but  equivalent  in  form  and  all  other  constants 
of  the  design  will  heat  to  a much  higher  temperature ; for, 
using  primed  symbols  to  represent  the  larger  transformer, 


A _ (Kvay*  _ ( Kva )-  _ _J_ 

A'  ~ ( K'vaf-  10*  (Kvaf-  3-2’ 

nnr1  kl  _ ( Kva)*  __  (Am)’  _ 1 

Kl  ( K'vcCy  10  *(Kvay  5-b 

whence  ^ = 1 : If* 

A A 


That  is,  the  full  load  losses  of  the  larger  transformer  are  about 
three  quarters  greater  per  square  inch  of  the  superficial  area  of 
the  transformer,  although  the  losses  are  only  about  half  as 
great  per  kilovolt-ampere  of  rated  capacity.  Had  the  size 
been  100  times  increased,  the  rate  of  liberation  of  heat  would 
have  been  over  200  per  cent  greater  per  square  inch  of  radiating 
surface.  Therefore,  transformers  of  large  size  require  either 
the  application  of  special  measures  for  carrying  away  the  heat 


MUTUAL  INDUCTION,  TRANSFORMERS 


541 


generated,  or  they  must  be  given  special  designs  which  would 
prove  bulky  and  expensive  in  sizes  of  50  Kva.  capacity  or 
over. 

Several  methods  for  cooling  are  in  use  : 

For  very  small  transformers  — a fraction  of  a kilowatt  capacity 
— no  special  arrangements  are  needed. 

For  transformers  from  a kilowatt  capacity  to  those  of  several 
hundred  kilowatts  capacity  the  most  common  method  is  to 
immerse  the  transformer  in  a tank  of  insulating  oil.  For  the 
smaller  sizes  this  is  not  necessary,  but  it  is  desirable,  as  the  oil 
tends  to  maintain  the  insulation  and 
close  up  punctures.  The  tanks  for 
smaller  sizes  of  transformers,  say  to 
about  50  Kva.  or  less  capacity,  are 
frequently  made  of  cast  iron  with  a 
smooth  outer  surface,  as  shown  in 
Fig.  47  ; while  those  of  larger  capacity 
are  made  of  cast  iron  or  riveted 
boiler  plate  with  a corrugated  surface 
in  order  to  present  more  radiating 
surface,  as  shown  in  Fig.  303.  The 
action  of  the  oil  is  to  circulate  up- 
ward through  ducts  in  the  warm  core 
and  over  the  surface  of  the  warm  coil 
sections  and  downward  alongthe  cooler 
interior  of  the  the  tank  or  case.  The 
coils  and  cores  are  so  arranged  that 
the  oil  can  readily  circulate  through 
them  ; thus  in  Fig.  298  the  cruciform 
shaped  core  leaves  triangular  channels 
through  which  the  oil  can  flow  between 
the  core  and  the  cylindrical  inner  face 
of  the  coil.  Where  necessary,  channels  or  ducts  are  left  between 
each  1|-  or  2^  inches  of  thickness  of  the  laminations.  In  trans- 
formers of  large  size  space  is  left  for  the  oil  to  circulate  freely 
between  sections  of  the  coils  ; thus  in  Fig.  302  is  shown  a trans- 
former intended  for  oil  cooling  in  which  the  coils  are  so 
separated. 

The  radiation  of  the  heat  from  the  case  is  commonly  at  the 
rate  of  one  watt  (joule  per  second)  to  each  4 to  8 square  inches 


Fig.  303.  — Transformer  Case 
having  a Corrugated  Surface 
for  increasing  the  Area  for  Heat 
Dissipation. 


542 


ALTERNATING  CURRENTS 


of  surface  when  the  case  temperature  is  from  35°  to  40°  centi- 
grade above  the  temperature  of  the  surrounding  air. 

In  the  case  of  large  transformers,  from  a couple  of  hundred 
to  several  thousand  kilowatts  capacity,  it  is  found  necessary  to 
artificially  cool  the  oil,  as  the  case  will  not  radiate  heat  to  the 
surrounding  air  fast  enough  to  keep  the  temperature  down  to  a 
reasonable  value.  In  this  case  it  is  common  to  circulate  water 
in  a spiral  of  pipes  located  inside  of  the  case  above  the  height 
of  the  transformer  proper.  Such  a transformer  is  said  to  be 
Water-cooled.  The  arrangement  is  illustrated  in  Fig.  304,  which 
represents  the  design  of  a water-cooled  transformer,  the  cooling 
coils  being  at  AA.  This  figure  shows  one  of  the  ways  for  mak- 
ing an  oil-proof  case  of  riveted  or  welded  boiler  plate,  which  in 
this  instance  does  not  need  to  be  corrugated  since  the  major  part 
of  the  cooling  is  accomplished  by  the  circulating  water  which 
flows  in  the  pipes.  The  figure  also  shows  means  which  may 
be  used  for  supporting  the  cores,  coils,  and  terminals,  and  for 
draining  the  oil  from  the  tank.  The  latter  is  accomplished  by  the 
pipe  containing  the  valve  B.  which  leads  to  a suitably  protected 
receiving  tank.  This  is  most  important  in  order  to  prevent  dis- 
astrous results  in  case  the  oil  in  the  transformer  catches  fire. 
Other  constructive  details  are  also  illustrated  in  the  figure.  The 
water  pipes  in  this  case  are  covered  with  cotton  tape  above 
the  oil  to  prevent  the  collection  and  drip  of  moisture. 

When  a water-cooled  transformer,  such  as  is  illustrated  in 
Fig.  304,  is  in  operation,  the  oil  in  contact  with  the  core  and 
windings  becomes  warm  and  tends  to  rise,  while  the  oil  cooled 
by  contact  with  the  cooling  pipes  tends  to  fall.  This  sets  up  a 
circulation  of  oil  upwards  through  the  ducts  and  over  the  sur- 
face of  the  transformer  into  contact  with  the  cooling  pipes,  then 
the  cooled  oil  falls  to  the  bottom  of  the  case  through  the  ample 
space  left  around  the  transformer.  In  this  manner  heat  is 
carried  away  from  the  transformer  and  delivered  to  the  cooling 
pipes.  The  cooling  water  may  be  caused  to  circulate  by  a 
pump  and  some  means  utilized  to  cool  it  after  its  exit  from  the 
cooling  coil.  The  water  may  thus  be  used  over  and  over.  For 
transformers  of  very  large  sizes  it  is  sometimes  desirable  to 
also  give  the  oil  a forced  circulation.  In  this  case,  oil  is 
drawn  from  the  transformer  by  means  of  a pump,  passed 
through  a surface  condenser  or  cooler  from  which  the  heat  is 


MUTUAL  INDUCTION,  TRANSFORMERS 


543 


abstracted  by  water,  and  returned  to  the  transformer  under 
pressure,  where  it  is  compelled  by  the  design  to  circulate 
through  the  core  and  windings  before  again  reaching  the  outlet. 


A good  method,  and  one  of  the  oldest,  of  cooling  indoor  trans- 
formers of  low  or  medium  high  voltage  is  by  means  of  a blast 


544 


ALTERNATING  CURRENTS 


of  air.  The  air  blast  is  produced  by  a fan,  and  directed  to  the 
transformer  by  way  of  an  air  chest  under  the  transformer  case, 
thence  through  the  core  and  windings,  and  thence  out  of  the 
top  cover  of  the  case.  The  windings  and  core  are  designed  with 
ducts  to  give  as  free  circulation  of  air  as  possible.  A large 
number  of  air-blast  transformers  can  be  located  over  one  air 
chamber  and  cooled  effectively  and  economically  in  this  way. 

Figure  305  shows  the  rise  of  temperature  of  the  conductors 
of  an  old  type  air-blast  transformer  of  200  kilowatts  capacity  as 
a function  of  the  period  of  operation,  operated  with  the  blast 


Fig.  305.  — Curves  showing  the  Effects  of  an  Air  Blast  in  Cooling  Transformers. 

and  without  the  blast.  Point  D shows  the  temperature  of  the 
stampings  after  the  transformer  has  been  operated  seven  hours 
with  blast  on;  curve  A shows  the  temperature  of  the  windings 
when  operated  at  full  load  without  blast;  curve  B shows  the 
temperature  of  the  windings  when  operated  at  full  load  with 
blast  of  1040  cubic  feet  per  minute;  and  curve  C shows  the 
temperature  of  the  air  issuing  from  the  transformer. 

The  insulation  in  transformers  is  composed  largely  of  the 
materials  described  for  generator  insulation,*  but  materials 
which  deteriorate  in  either  hot  or  cold  oil  cannot  be  used  in 
oil-cooled  or  water-cooled  transformers.  The  usual  substances 
used  for  the  insulation  between  the  cores  and  coils  are  mica  and 

* Art.  36. 


MUTUAL  INDUCTION,  TRANSFORMERS 


545 


varnished  cotton  cloth  or  paper,  with  the  addition  of  blocks  of 
wood  or  press  board  where  large  volumes  of  insulating  materials 
are  demanded.  The  wire  is  usually  double  cotton  covered,  and 
varnished  cloth  is  used  between  the  layers.  The  voltage 
between  layers  seldom  exceeds  200  volts.  In  fact,  200  volts 
per  layer  is  high.  Each  winding  is  divided  into  thin  sections, 
giving  a total  of  not  over  3000  or  4000  volts,  which  figures  are 
also  rather  high  for  the  best  construction.  These  coil  sections 
are  thoroughly  impregnated,  usually  by  using  a vacuum  process, 
with  linseed  oil,  asphaltum,  or  compounds  derived  from  coal 
tar.  It  is  evident  that  in  transformers  for  high  voltages,  such 
as  from  60,000  to  100,000  or  more  volts,  the  insulation  between 
the  cores  and  coils  must  have  very  high  dielectric  strength  so 
that  much  mica  is  employed.  The  insulation  between  coil  sec- 
tions is  largely  accomplished  by  the  cooling  oil  which  surrounds 
them,  as  the  sections  are  kept  apart  by  narrow  separators  of 
wood  or  pressboard. 

The  terminals  of  a transformer  are  troublesome  to  insulate, 
as  they  are  not  in  oil,  and  especially  since  high  disruptive  forces 
are  apt  to  be  set  up  at  those  points  upon  opening  or  closing 
circuits.  Fig.  306  shows  at  A a pair  of  66,000  volt  leads  for 
a transformer  located  under  roof.  The 
wires  themselves  are  insulated  to  many 
times  their  diameter  and  are  bushed  in 
porcelain  where  they  pass  through  the 
iron  cover  plate  of  the  transformer. 

At  B is  shown  a lead,  for  a similar 
voltage,  to  be  used  in  a transformer 
intended  for  use  out  of  doors.  The 
multiple  petticoated  porcelain  insulator 
is  for  the  purpose  of  effectively  disposing 
of  rain  water. 

A desirable  form  of  terminal  is  now 
manufactured  with  alternate  layers  of 
insulating  material  and  tin  foil,  the 
function  of  the  latter  being  to  distribute  the  electric  stress  uni- 
formly through  the  dielectric.* 

The  oil  in  transformers  should  withstand  a breakdown  test 
of  about  30,000  volts  when  the  voltage  is  applied  between 

* Trans.  A.  I.  E.  E.,  Vol.  28,  p.  209. 

2 N 


Fig.  306.  — Terminals  for 
06,000  Volt  Transformer. 
A,  for  Indoor  Service.  B, 
for  Outdoor  Service. 


546 


ALTERNATING  CURRENTS 


PRIMARY  SIDb 


SECONDARY  SIDE 


electrodes  having  spherical  ends  of  .5  centimeter  diameter 
placed  .4  centimeter  apart.  It  should  be  free  from  moisture 
and  other  foreign  substances,  such  as  materials  containing 
sulphur,  alkali,  or  acids;  should  be  a mineral  oil  manufactured 
by  the  fractional  distillation  of  petroleum;  and  should  be  very 
fluid  and  neither  solidify  or  form  wax  at  0°  centigrade.  Its 
flash  point  should  be  at  least  170°  centigrade.*  The  removal 
of  moisture  which  may  have  got  into  otherwise  satisfactory  oil 
may  be  accomplished  by  filtering  the  oil  through  common  soft 
blotting  paper.  It  may  also  be  accomplished  by  heating  the 
oil  to  a temperature  above  100°  centigrade  and  stirring  it. 

140.  Methods  of  connecting  Constant  Voltage  Transformers 
and  some  Features  of  their  Operation.  — Single-phase  trans- 
formers can  be  connected  to  single  phase  circuits  either  to  raise 
or  lower  the  voltage,  as  shown  by  the  upper  or  lower  trans- 
formers in  the  dia- 
gram of  Fig.  307. 
If  the  three-wire 
system  is  used  in 
the  secondary  cir- 
cuit, two  trans- 
formers in  series 
can  be  used,  but 

more  commonly  the  neutral  wire  is  tapped  to  the  middle  of 
the  secondary  winding,  as  shown  by  the  dotted  line  in  Fig. 
307.  Lighting  transformers  generally  have  their  secondary 
windings  divided  into  two  coils  with  their  terminals  brought 
out  to  terminal  blocks  so  that  the  three-wire  connection  can 
be  readily  made. 

Two  or  more  transformers  are  sometimes  connected  with 
their  secondary  windings  in  parallel  on  the  same  secondary  cir- 
cuit ; in  which  case  it  is  necessary  that  they  shall  have  equal 
ratios  of  transformation,  and  that  their  internal  resistances  and 
reactances  shall  be  inversely  proportional  to  their  respective  rated 
capacities,  or  they  will  not  divide  the  total  load  proportionally 
to  their  rated  capacities.  Thus,  consider  two  transformers 
designed  for  equal  primary  and  no-load  secondary  voltages  and 
equal  rated  capacities,  but  having  their  total  internal  impedances, 
resolved  in  terms  of  secondary  circuit  equivalents,  in  the  ratio 
* U.  S.  Government  Specifications. 


Fig.  30T.  — Diagram  showing  Connections  of  Transformers. 


MUTUAL  INDUCTION,  TRANSFORMERS 


547 


of  the  vectors  Zt  and  AZt , where  A is  a constant,  the  trans- 
formers being  connected  to  the  same  primary  circuit.  The 

currents  in  the  two  are  respectively  proportional  to  ~ and  — . 

Zt  AZt 


The  load  on  one  transformer  is  therefore  less  than  that  on  the 
other  transformer  in  the  ratio  of  Zt  : AZt  = 1 : A.  Moreover, 
the  secondary  terminal  voltages  may  not  be  exactly  in  the  same 
phase,  and  an  undesirable  cross  current  will  flow,  so  that  the 
arithmetical  sum  of  the  currents  carried  by  the  two  transformers 

X 

is  greater  than  the  current  in  the  load,  unless  — ^ is  equal  for  the 

tit 


two  transformers. 

If  the  ratios  of  transformation  on  open  circuit  are  different 
in  the  ratio  of  E.2  + B to  Ev  where  E2  is  the  secondary  ter- 
minal voltage  of  one  transformer,  the  transformer  having  the 
lower  secondary  voltage  absorbs  power  from  the  other  until  the 
total  load  becomes  great  enough  to  cause  the  terminal  voltage 
of  the  latter  to  drop  to  E2  volts.  With  further  increase  of 
load  the  first  continues  to  carry  the  load  required  to  bring  its 
voltage  to  E2  and  divides  further  increases  with  the  second 
transformer  in  the  proportion  stated  in  the  preceding  paragraph. 
If  the  internal  impedances  are  both  very  low  compared  to  the 
load  impedance,  differences  in  the  division  of  load  are  little 
noticeable,  but  it  is  always  wise  to  keep  these  conditions  in 
mind  when  designing  transformers  for  manufacture  or  selecting 
them  for  use. 

When  secondary  circuits  of  transformers  fed  from  one  primary 
circuit  are  to  be  connected  in  parallel  it  is,  of  course,  essential 
that  they  be  connected  in  the  proper  relation  so  that  they 
will  not  short-circuit  upon  each  other.  It  is  thus  necessary  to 
know  or  determine  the  Polarity  of  the  windings.  The  relative 
polarity  of  two  transformers  may  be  obtained  by  connecting  up 
one  transformer,  and  then,  after  attaching  the  primary  terminals 
of  the  second  to  the  primary  line,  connecting  one  of  its  second- 
ary terminals  directly  to  a secondary  line  conductor  and  the 
other  secondary  terminal  to  the  other  secondary  line  conductor 
through  a voltmeter.  If  the  transformers  are  connected  in 
series  instead  of  parallel  relation,  the  voltmeter  will  read 
double  the  normal  secondary  voltage.  A light  fuse  may  be 
used  instead  of  the  voltmeter,  and  the  blowing  of  the  fuse  will 


548 


ALTERNATING  CURRENTS 


denote  wrong  connections.  Figure  807  a shows  correct  connec- 
tions for  two  transformers  in  parallel.  Figure  307  b (1  and  2) 

show  incorrect  connec- 
tions for  parallel  opera- 
tion, as  the  transformers 
in  each  diagram  are 
short-circuited  on  each 
other.  Figure  30  7 c 
shows  the  connections 
whereby  a three-wire, 
may  be  obtained  by  means 
two-wire  primary  circuit 


Fig.  307  a.  — Correct  Connections  for  Two 
Transformers  in  Parallel. 


circuit 
from  a 


1 

1 

CD 


single-phase,  secondary 
of  two  transformers 
when  a middle  con- 
nection to  the  second- 
ary winding  is  not  avail- 
able. 

The  transformation 
of  voltage  in  polyphase 
circuits  may  be  com- 
passed by  using  single- 
phase transformers  in 
groups.  A quarter- 
phase  circuit  then  re- 
quires two  transformers 
at  each  point  of  trans- 
formation, and  a tri- 
phase circuit  requires 
either  two  or  three 
transformers.  The  individual  transformers  must  each  have  a 
capacit)'  equal  to  the  power  required  to  be  transformed  by 

it  divided  by  the 
power  factor  of  the 
part  which  it  sup- 
plies. As  the  power 
factor  of  an  incan- 
descent lamp  load 
by  is  practically  unity, 
and  since  circuits 
supplying  induction  motors  are  likely^  to  have  full  load  power 
factors  not  higher  than  0.8,  and  further  since  when  such  motors 


Fig.  307  b. — Incorrect  Connections  for  Two  Trans- 
formers in  Parallel. 


Fig. 


307  c.  — Three-wire  Secondary  Circuit  supplied 
Two  Transformers. 


MUTUAL  INDUCTION,  TRANSFORMERS 


549 


are  only  partly  loaded  the  power  factor  is  much  decreased,  it  is 
evident  that  transformers  which  supply  currents  to  motors  must 
be  of  greater  kilovolt-ampere  capacity  than  those  which  supply 
equal  power  to  incandescent  lamps.  This  rule  applies  equally  to 
single-phase  and  polyphase  circuits  and  it  is  important  to  bear  in 
mind  when  purchasing  transformers  that  their  capacities  should 


Fig.  308.  — Diagram  of  Transformer  Connections  for  a Three-wire 
Quarter-phase  System. 


GROUP  1 


be  suitable  for  the  kilovolt  amperes  of  the  load  to  which  they  are 
to  be  connected.  Figure  308  represents  the  connections  for  a 
quarter-phase  circuit  when  a common  return  wire  is  used  in 
both  primary  and  secondary  circuits.  In  the  case  of  a four- 
wire  quarter-phase  circuit,  the  transformers  may  be  kept  inde- 
pendent, one  single-phase  transformer  ABC  — 
being  provided  for  each  phase  of  the 
circuit.  A four-wire  circuit  may  be 
used  for  either  the  primary  or  second- 
ary circuit  and  a circuit  with  common 
return  for  the  other  by  associating 
the  four-wire  connection  with  one 
side  of  two  transformers  and  the 
three-wire  connection  of  Fig\  308 
with  the  other  side  of  the  trans- 
formers. In  Fig.  309  are  shown  the 
usual  connections  utilized  for  groups 
of  three  single-phase  transformers 
for  three-phase  transformation.  The 
figure  shows  three  groups  each  of 
which  comprises  three  transformers. 

In  Group  1 the  primary  windings  ABC' 
and  the  secondary  windings  of  the  Fig.  309.  — Diagram  showing 

transformers  are  both  connected  in  Ways  of  Connecting  ihree 

Single-phase  Transformers  to 

A arrangement.  In  Group  2 the  Tri-phase  Lines. 


GROUP  2 


GROUP  3 


550 


ALTERNATING  CURRENTS 


primary  windings  and  the  secondary  windings  of  the  trans- 
formers are  both  connected  in  Y arrangement.  In  Group  3 the 
primary  windings  of  the  transformers  are  connected  in  A ar- 
rangement and  the  secondary  windings  are  connected  in  Y 
arrangement.  The  third  arrangement  may  be  reversed  so  as  to 
connect  the  primary  windings  in  Y and  the  secondary  windings 
in  A. 

When  transformers  are  to  be  connected  in  any  of  these 
arrangements,  the  terminals  must  be  joined  correctly  according 
to  the  polarities,  or  the  voltage  phases  will  be  unbalanced  in 
the  secondary  circuit.  Thus,  when  the  secondary  coils  are 
properly  connected  and  the  system  is  balanced  as  to  the  vol- 
tages. the  three  coil-voltages  may  be  laid  out  graphically  as  a sym- 
metrical phase  diagram, 
as  shown  by  the  vectors 
represented  by  OA,  OB , 
and  0(7,  in  Fig.  310. 

By  reversing  the  con- 
nections of  one  of  the 
windings  of  one  of  the 
transformers  in  a Y ar- 
rangement, its  secondary 
voltage  is  reversed.  For 
instance,  reversing  the 
transformer  giving  the 
voltage  OB  of  Fig.  310 
gives  the  voltage  vector 
OJD  of  the  diagram,  which, 
using  the  relations  shown  in  the  figure,  lies  60°  behind  OA 
and  60°  ahead  of  0(7  instead  of  in  the  proper  position  120° 
ahead  of  OA  and  behind  0(7.  The  voltages  between  the  sec- 
ondary line  wires  A and  B , B and  C,  and  (7  and  A are  there- 
fore changed  to  the  values  and  vector  relations  shown  by  the 
lines  AT),  DC , and  CA  in  the  vector  triangle  ADC , instead 
of  their  balanced  relations  for  the  proper  connections  shown 
by  A B,  B (7,  and  CA.  The  diagram  shows  by  inspection  that 
the  voltages  AD  and  DC  are  numerically  equal  to  the  wye 
voltage  CO  of  the  circuit,  but  CA  is  equal  to  V3  times  the 
wye  voltage.  The  angles  between  AD , DC , and  CA  are  re- 
spectively equal  to  60°,  150°  and  150°. 


Fig.  310.  — Diagram  of  y-connected  Transformer 
Voltages  in  a Tri-phase  Circuit  to  show  the  Effect 
of  reversing  the  Connections  of  One  Winding. 


MUTUAL  INDUCTION,  TRANSFORMERS 


551 


C B' 

Fig.  310  a.  — Diagram  of  A-connected  Transformer 
Voltages  in  a Tri-phase  Circuit  to  show  the  Effect 
of  reversing  the  Connection  of  One  Winding. 


When  the  secondary  coils  are  connected  in  delta,  the  diagram 
of  Fig.  310  a can  be  applied,  but  now  OA,  OB , and  OC  repre- 
sent, not  only  the  coil  voltages,  but  also  the  line  voltages. 
Hence,  when  the  con- 
nection of  the  wind- 
ing of  transformer 
B is  reversed,  the 
voltage  between  B 
and  C is  reversed  and 
the  vector  diagram 
of  voltages  AB , B'  C 
( = — BO)  and  OA 
does  not  close.  This 
leaves  an  uncompen- 
sated voltage  B' B to  make  a circulating  current  in  the  mesh, 
which  would  burn  out  the  transformers. 

In  a wrongly  connected  wye  system,  unbalanced  voltages  are 
impressed  on  the  secondary  circuit  which  would  ordinarily  lead 
to  the  flow  of  unbalanced  currents  in  both  primary  and  secondary 
circuits,  and  if  maintained  would  cause  inconvenience  to 
users  of  apparatus  of  which  the  load  might  be  composed. 
In  a wrongly  connected  delta  system  the  transformers  are  jeop- 
ardized by  a circulating  current  caused  by  the  resultant  vol- 
tage in  the  mesh.  This  is  generally  obviated  by  use  of  proper 
fuses  and  other  circuit  breakers,  but  it  is  well  to  guard  against 
such  dangers  by  making  polarity  tests  before  cutting  the  trans- 
formers into  circuit. 

It  must  be  observed  that  the  phases  of  the  voltages  between 
three-phase  secondary  line  wires  are  opposite  to  the  voltages 
impressed  between  the  primary  line  wires  when  both  primary 
and  secondary  windings  of  the  transformers  are  connected  both 
either  in  delta  or  in  wye  arrangement.  This  is  not  true,  how- 
ever, when  the  primary  windings  are  connected  in  one  arrange- 
ment and  the  secondary  windings  are  connected  in  the  other 
arrangement,  as  this  results  in  a displacement  of  the  secondary 
voltage  phase  30°  from  the  position  of  opposition,  for  the  reasons 
heretofore  explained.*  The  line  voltages  are  shifted  forward 
30°  when  going  through  the  transformers  from  delta  to  wye 
and  are  shifted  backward  30°  when  going  through  the  trans- 

* Art.  100. 


552 


ALTERNATING  CU RR ENTS 


formers  from  wye  to  delta.  For  this  reason  groups  of  three 
transformers  operating  in  three-phase  circuits  cannot  be  con- 
nected in  parallel  on  both  the  primary  and  secondary  circuits 
unless  due  consideration  is  given  to  the  manner  of  their  con- 
nections. Transformers  connected  delta  primary  and  delta 
secondary  and  transformers  connected  wye  primary  and  wye 
secondary  can  be  paralleled  on  the  secondary  side  with  like 
groups  and  with  each  other,  when  operated  from  the  same  pri- 
mary circuit;  but  they  cannot  be  paralleled  with  transformers 
connected  delta  primary  and  wye  secondary  nor  with  trans- 
formers connected  wye  primary  and  delta  secondary.  Each 
of  the  two  last-named  arrangements  may  be  paralleled  with 
like  groups,  but  neither  may  be  paralleled  with  the  other. 

When  the  connection  of  the  windings  is  delta  primary  and 
delta  secondary  or  wye  primary  and  wye  secondary,  it  is  mani- 
fest that  the  ratio  of  primary  and  secondary  voltages  at  no  load 
is  equal  to  the  transformation  ratio  of  the  individual  trans- 
formers, but  tins  is  not  true  when  the  connections  of  primary  and 
secondary  windings  differ.  When  the  arrangement  is  delta 
primary  and  wye  secondary,  the  wye  voltage  of  the  secondary 
circuit  is  equal  to  the  primary  impressed  voltage  divided  by  the 
ratio  of  transformation,  and  in  a balanced  circuit  the  secondary 
line  voltage  is  V3  times  as  great.  When  the  arrangement  is 
Avye  primary  and  delta  secondary,  the  secondary  line  voltage 
is  equal  to  the  primary  wye  voltage  divided  by  the  ratio  of 
transformation.  The  primary  line  voltage  is  V3  times  the  pri- 
mary wye  voltage,  and  the  secondary  line  voltage  is  therefore 
1/V3  as  great  in  a balanced  circuit  as  would  be  given  by  divid- 
ing the  primary  line  voltage  by  the  ratio  of  transformation  of 
the  transformers. 

In  Fig.  311  is  shown  a diagram  for  connecting  two  trans- 
formers on  a tri-pliase  circuit  in  what  is  sometimes  called  the 
Open  delta  or  Vee  connection.  The  two  arrangements  shown 
are  alike  in  result,  but  the  transformers  are  attached  to  different 
phases.  The  secondary  voltages  of  the  respective  transformers 
are  represented  in  Fig.  312  by  the  lines  AB  and  CA<  cor- 
responding with  the  connection  shown  in  the  upper  part  of  the 
figure,  while  the  distance  BO  represents  the  resulting  secondary 
line  voltage  that  is  maintained  jointly  by  the  two  transformers 
between  the  wires  B and  C.  When  three  transformers  are  con- 


MUTUAL  INDUCTION,  TRANSFORMERS 


553 


nectecl  in  delta  arrangement  and  one  becomes  disabled  so  as  to 
be  disconnected  from  the  circuit,  a large  part  of  the  full  load 
can  still  be  carried  by  the  two  transformers  a b C 
that  remain  in  V connection.  With  three 
transformers  connected  in  the  delta  arrange- 
ments the  total  power  delivered  to  the  cir- 
cuit is  equal  to  V3  cos  #2,  and  each 
transformer  delivers  one  third  of  this  power, 

E„Ir,  COS  Or, 


or 


2 2 


V3 


After  the  circuit  of  one  of 


the  three  transformers  has  been  opened,  as- 
suming the  load  to  remain  the  same,  each 
of  the  remaining  transformers  must  deliver 

VBAJgig  cos  02  ^ to  the  receiving  circuit. 

Since  the  current  in  each  line  wire  remains 
unchanged  as  does  the  voltage  between  line 
wires,  it  is  evident  that  in  the  transformer 
coils  there  will  be  additional  angular  dis- 
placement corresponding  to  an  angle  of 
v/3 

cos  _1- — , that  is,  ± 30°.  The  total  phase  dis- 


Fig.  311.  — Connections 
for  using  Two  Trans- 
formers on  a Tri- 
phase Circuit.  V Con- 
nection. 


SC' 


placement  in  the  two  transformers  is  then  re- 
spectively #2  + 30°  and  0 2 — 30°.  From  this  it  will  be  seen  that 
the  current  in  each  transformer  increases  in  a greater  proportion 
than  does  the  power  delivered  by  the  trans- 
former. Hence  when  two  transformers  are  op- 
erating in  open  delta  (i.e.  V connection)  on  a 
tri-phase  circuit  the  transformers  will  not 
(without  being  overloaded)  deliver  their  full 
rated  capacity  in  kilowatts  even  when  supply- 
ing power  to  a non-reactive  load.  Thus,  three 
transformers  each  of  200  Ivva.  rated  full  load 
capacity  being  delta  connected  to  a tri-phase 
showing  Voltages  circuit  and  delivering  a total  of  600  kilowatts 

of  Secondary  to  a balanced  receiving'  circuit  having  unity 
Phases  of  a Tri-  ° J 

phase  System  when  Power  factor,  when  one  of  these  transformers 

Two  Transformers  has  its  secondary  circuit  opened  by  accident 
are  used,  and  the  eacp  0f  tq-,e  two  remaining  transformers  will 
Wrong  Connections,  deliver  300  kilowatts  to  the  receiving  circuit, 


554 


ALTERNATING  CURRENTS 


but  the  current  in  each  transformer  will  be  out  of  phase 
with  its  secondary  voltage  by  an  angle  of  30° ; hence,  the  kilo- 
volt amperes  delivered  by  each  transformer  must  be  equal  to 

V3 

300  h — — = 346  Ivva.  and  the  transformers  are  73  per  cent  over- 
loaded. This  is  but  one  of  many  possible  transformer  combi- 
nations which  result  in  a phase  displacement  between  voltage 
and  current  in  the  transformers  ; and  in  every  case  when  such 
a combination  is  made  the  kilowatt  output  of  the  transformers 
is  less  than  the  sum  of  the  kilovolt  amperes  on  account  of  the 
internal  phase  displacement  of  the  current. 

If  the  connection  of  either  the  primary  or  the  secondary 
winding  of  the  transformer  giving  voltage  CA  in  Fig.  312  is 
reversed,  the  three  secondary  line  voltages  become  AB , BC, 
and  C'A.  The  last  is  equal  to  — CA.  This  is  a workable  com- 
bination because  the  third  side  of  the  vector  diagram  BC  is  not 
fixed  in  position  or  magnitude  by  its  own  transformer,  as  it  is 
in  the  case  of  delta  connection  of  three  transformers  ; but  it 
gives  unbalanced  secondary  voltages.  The  diagram  shows  by 
inspection  that  the  voltage  BC  is  numerically  V3  times  vol- 
tages AB  and  C'A,  and  the  angles  between  the  voltages  AB , 
BC , and  CA  are  150°,  150°,  and  60°.  This  arrangement  is 
sometimes  used  to  get  a 60°  phase  difference. 

A substitute  for  the  V connection  may  be  made  with  two 
transformers,  in  which  the  windings  of 


Ci 


0 


B, 


Ax 


a/vvWa  Wvw\ 


B,. 


one  transformer  possess  ^ times  as  many 

turns  as  the  windings  of  the  other.  In 
this  case,  one  end  of  each  winding  of  the 
former  is  connected  to  the  middle  of  the 
corresponding  winding  of  the  latter,  as 
illustrated  in  Fig.  313.  This  is  called 
the  Tee  connection  of  transformers.  The 
windings  of  fewer  turns  which  are  pos- 
sessed by  one  transformer  are  sometimes 
called  Teaser  windings.  The  potential  of 
Av  the  line  end  of  the  primary  teaser 
winding,  is  fixed  by  the  tri-phase  relations 
of  the  line  voltages,  while  the  potential  of  the  other  end  is  fixed 
by  being  midway  between  the  potentials  of  Bx  and  Cv  by  reason 


) \ 

LOAD  / ' 


Fig.  313.  — Tee  Connection 
of  Transformers  for  Tri- 
phase Circuit. 


MUTUAL  INDUCTION,  TRANSFORMERS 


555 


of  the  connection  of  the  teaser  at  the  mid-point  0.  The  vector 
diagram  of  voltages  is  given  in  Fig.  314,  in  which  BO  is  the 
voltage  from  Bx  to  0 of  Fig.  313,  00  is  the  voltage  from  0 to 
Ov  and  OA  is  the  voltage  from  0 to  Av  The  line  voltages  are 
represented  by  AB , BO,  and  CA.  The  tensor  value  of  voltage 


V3 

OA  is  — - times  that  of  either  line  voltage.  The  voltages  of 
2 


the  secondary  circuit  are  proportional  to  those  of  the  primary 
circuit,  by  reason  of  the  correspondence  of  the  pri-  a 

mary  and  secondary  windings  and,  therefore,  are 
also  represented  by  the  vectors  of  Fig.  314.  That 
is,  the  secondary  line  voltages  must  be  proportional 
in  magnitude  and  angular  relations  to  the  vectors 
represented  by  AB,  BO,  and  OA  of  the  figure. 

Assuming  the  system  to  be  balanced,  the  line  FlgGranfo{  v7itages 
currents  flowing  in  the  two  halves  of  the  coil  produced  by 
B101  or  B2C2  are  numerically  equal  to  each  other  arranged0 in* Tee 
and  are  120°  apart  in  phase.  At  0 they  add  Connection, 
together  vectorially  to  form  the  line  current  which  flows 
through  the  teaser,  which  is  numerically  equal  to  each  of  the 
others  and  120°  in  advance  of  one  and  120°  behind  the  other, 
in  accordance  with  the  law  that  the  vector  sum  of  the  currents 
at  a junction  must  be  zero.  The  kilovolt-auqjeres  capacity  of 
the  transformers  manifestly  must  be  greater  than  the  kilowatt 
output  even  when  the  load  is  non-reactive. 

The  T connection  of  two  transformers  for  tri-phase  circuits 
possesses  the  disadvantage,  compared  with  the  V connection,  that 
the  two  transformers  cannot  be  alike  unless  part  of  the  winding 
of  one  is  allowed  to  be  idle  and  its  capacity  is  therefore  sacri- 
ficed ; but  the  T connection  has  a countervailing  advantage 
in  the  fact  that  the  system  is  not  so  likely  to  become  unbalanced 
as  when  the  V connection  is  used. 

The  relative  transformer  capacities  required  when  the  delta, 
wye,  vee,  and  tee  connections  are  used  for  three-phase  to  three- 
phase  transformation  may  be  determined  by  the  following  rela- 
tions: For  delta  and  Avye,  the  capacity  of  each  of  the  three 

transformers  must  be  equal  to  one  third  of  the  total  load  in 
kilovolt  amperes.  This,  for  any  power  factor,  is  equal  to  Erl 


= EI±  = — - El  (where  Er,  E,  I,  and  are  the  wye  voltage, 

V3 


55G 


ALTERNATING  CURRENTS 


line  voltage,  line  current,  and  delta  current  of  the  circuit),  for 
in  the  delta  the  primary  coil  current  equals  — ^ I and  in  the 

wye  the  primary  coil  voltage  equals  E.  The  three  required 

Vb 

transformers  then  have  a combined  kilovolt  ampere  capacity 
of  V3  El.  If  the  windings  are  delta-connected,  they  must  be 
designed  to  produce  full  line  voltage  and  carry  a current  equal 
to  the  line  current  divided  by  V3,  while  if  the  windings  are 
wye-connected  they  must  be  designed  to  produce  a voltage 
equal  to  line  voltage  divided  by  V3  and  to  carry  full  line  cur- 
rent. In  the  tee  connection  the  coils  carry  the  same  currents 
as  the  line  wires,  but  the  voltage  of  one  transformer  having  the 


teaser  winding  is  onlv 


V3 


times  as  large  as  the  line  voltage. 


to  El , while  that  of  the  main  transformer  must  be  equal  to 


The  kilovolt  ampere  capacity  of  the  teaser  transformer  is  equal 
V3 
2~ 

El.  The  combined  capacity  of  the  two  transformers  required 
for  an  output  of  V3  El  kilovolt  amperes  is  therefore  — El 


In  the  vee  connection  the  voltage  in  the  primary  windings  of 
each  of  the  two  transformers  is  E and  the  current  I.  The  cur- 
rents entering  the  windings  at  the  outer  ends  of  the  vee  are 
equal  to  the  two  corresponding  line  currents,  and  these  vectori- 
ally  add  at  the  apex  at  a phase  angle  of  120°,  which  gives  an 
equal  current  for  the  third  wire.  Hence  the  combined  capacity 
of  the  two  transformers  should  be  2 El. 

It  should  be  understood  that  in  the  case  of  the  tee  connection 
or  the  vee  connection  the  relations  of  the  currents  and  voltages 
in  the  transformers  so  connected  are  not  necessarily  the  same 
as  the  phase  relations  between  line  currents  and  line  voltages. 
In  the  case  of  the  tee  connection  the  current  in  the  teaser 
winding  has  the  same  angular  relation  to  the  voltage  of  that 
winding  as  the  line  current  lias  to  the  line  voltage,  but  as  pre- 
viously explained  in  this  Article  there  is  an  inherent  phase 
displacement  of  the  currents  in  the  two  halves  of  the  main 
transformer  winding  irrespective  of  the  power  factor  of  the 
line.  In  the  case  of  the  vee  connection  the  currents  in  both 
transformers  are  out  of  phase  with  the  respective  transformer 


MUTUAL  INDUCTION,  TRANSFORMERS 


557 


voltages  as  explained  earlier.  Because  of  this  condition  the 
total  kilovolt  ampere  capacity  of  the  transformers  connected 
in  tee  and  in  vee  must  be  in  excess  of  the  total  kilovolt  amperes 
supplied  by  the  line  or  to  the  load. 

When  the  three  transformers  are  connected  in  wye  in  a tri- 
phase system,  a fourth  wire  may  be  connected  to  the  neutral 
point  or  common  junction  point,  and  part  or  all  of  the  load 
may  be  connected  from  the  three  independent  wires  to  the 
neutral  wire.*  This  reduces  the  load  voltage,  as  shown  in  Fig. 
315,  to  the  secondary  coil  voltage,  while  maintaining  the  voltage 
between  the  three-phase  wires  at  V3  times  that  value.  When 
the  load  is  not  perfectly  balanced,  currents  will  flow  in  the 
neutral  wire,  and  it  is  usual  to  make  the  neutral  conductor  of 


! S 

K 

E., 

j 

f» 

:i 

■ n 

/ 

i 

E„ 

r 

i i 



e2 

vf 

Fig.  315  — Tri-phase  Transformer  Connections  when  a Neutral  Wire  is  used. 

the  same  size  as  the  other  three ; but  as  the  phase  voltages 
are  equal  to  V3  times  the  load  voltages,  and  as  the  weight  of 
copper  is  inversely  as  the  square  of  the  voltage,  the  weight 
of  copper  required  for  the  transmission  of  a given  power  over 
a given  distance  with  a fixed  loss  of  power  is  only  ^ as  great  as 
when  the  ordinary  delta  or  wye  arrangement  is  utilized  with 
the  same  voltage  on  the  load,  and  these  in  turn  require  only 
f as  great  a weight  of  copper  as  a single-phase  or  a four-wire 
quarter-phase  system  with  the  same  voltage  between  the  con- 
ductors of  each  phase.  Consequently,  the  arrangement  of  Fig. 
315  requires  only  ^ as  great  a weight  of  copper  as  a two-wire 
single-phase  or  four-wire  two-phase  circuit  with  the  same  vol- 
tage on  the  load. 

The  neutral  point  of  a tri-phase  system  is  likely  to  float  from 
the  balanced  center  when  the  phases  become  unbalanced.! 
That  is,  the  voltage  between  each  of  the  line  wires  and  the 
* Art.  100.  t Art.  103.  Examples. 


558 


ALTERNATING  CURRENTS 


neutral  wire  is  dependent  upon  the  character  of  the  load  in  the 
several  phases. 

It  is  common  practice  to  ground  the  neutral  point  of  wye  con- 
nected secondary  circuits,  both  for  the  purpose  of  reducing  the 
tendency  of  unbalancing  and  of  reducing  danger  which  might 
threaten  life  or  property  through  having  exaggerated  voltages 
arising  for  any  reason  between  the  line  conductors  and  ground. 
Thus,  for  instance,  if  a single  phase  of  a high  voltage  primary 
circuit  accidentally  makes  connection  into  the  secondary  wind- 
ings where  the  neutral  is  grounded,  it  may  be  brought  to  earth 
potential,  and  it  cannot  rise  with  reference  to  the  ground  poten- 
tial above  the  potential  of  the  secondary  phase  wire  with  which 
it  comes  in  contact.  The  neutral  conductors  of  single-phase 
three-wire  secondary  circuits  are  commonly  grounded  to  ob- 
tain similar  protection.  Variations  of  voltage  between  earth 
and  the  line  wires  of  an  insulated  system  may  also  occur  from 
other  causes  than  the  illustration  just  given,  but  when  a system 
of  conductors  has  a ground  connection,  whether  from  a wye 
neutral  or  otherwise,  the  voltages  between  the  ground  and  the 
several  line  wires  evidently  depend  upon  the  values  of  the  vol- 
tages between  the  wires  themselves.  Thus,  if  one  line  wire  of  a 
wye  or  delta  load  is  connected  to  ground,  the  voltages  between 
the  ground  and  the  other  line  wires  are  equal  to  the  full  line 
voltage.  When  the  wye  neutral  point  is  grounded,  the  voltage 
between  either  phase  wire  and  ground  is  equal  to  the  voltage  be- 
tween the  neutral  point  and  the  phase  wire,  which,  in  a balanced 
three-phase  system,  is  equal  to  the  line  voltage  divided  by  V8. 

When  the  neutral  points  of  several  generators  or  transform- 
ers are  brought  into  conjunction  as  by  grounding,  any  third  har- 
monics of  the  voltages  induced  in  the  windings  may  cause  third 
harmonic  cross  currents  to  flow  through  the  ground  connections, 
windings,  and  line  wires,  which  currents  could  not  exist  except  for 
such  connections.  This  condition  may  be  controlled  by  inserting 
a small  amount  of  resistance  in  each  conductor  leading  to  ground. 

Although  grounding  at  special  points  may  be  made  designedly 
with  good  effect,  accidental  grounding  of  otherwise  ungrounded 
systems  is  generally  objectionable,  as  it  ma3r  be  the  cause  of 
serious  disturbances.  This  is  particularly  true  of  systems  of 
very  high  voltage.  Even  though  this  relation  may  not  prove 
serious  at  the  point  where  accidental  grounding  occurs,  the 


MUTUAL  INDUCTION,  TRANSFORMERS 


559 


effect  is  carried  more  or  less  directly,  depending  upon  the  com- 
bination of  wye  and  delta  connections  of  the  various  step-up  and 
step-down  transformers,  over  the  whole  distribution  system  and 
may  at  one  point  or  another  be  the  cause  of  accident.  It  is 
essential,  then,  that  ground  detectors  be  used  on  all  insulated 
portions  of  a distribution  system,  and  that  when  grounding  is 
desired  the  ground  connection  be  made  with  due  reference  to 
the  voltages  and  insulation  strength  over  the  whole  system. 

Disastrously  high  voltages  may  be  caused  under  certain  con- 
ditions when  the  primary  circuit  of  a transformer,  having  its 
secondary  winding  connected  into  a long  high  voltage  three- 
phase  transmission  line  of  large  electrostatic  capacity,  is  open 
circuited.  Thus,  suppose  a group  of  three  transformers  are  con- 
nected in  wye  fashion  to  the  primary  and  secondary  conductors 
of  a transmission  system,  as  illustrated  in  Fig  316.  If  the  pri- 
mary coil  connected  into  lead  A1  is  open-circuited  for  any  rea- 
son, and  the  sec- 
ondary coil  remains 
connected  into  its 
circuit,  the  voltage 
between  wires  B2 
and  C2  may  not  be 
seriously  affected, 
but  the  voltages  be- 
tween A0  and  B, 


Aj  A 


Cl 

Tc2 

Bf 

L 

2’  Fig.  316.  — Diagram  showing  Three  Transformers  Con- 


and  C2  and  A2,  may 
become  dangerously 
high  ; for  now  the 
coil  02A2  acts  as  a high  inductive  reactance  which  is  in  series 


nected  in  Wye  Fashion  in  a Tri-phase  System,  with  the 
Primary  Circuit  of  One  Transformer  Open.  Illustration 
of  a Dangerous  Condition. 


with  what  may  happen  to  be  an  equivalently  high  capacity  re- 
actance of  the  line  conductor  leading  from  A2  to  the  load.  This 
would  produce  a condition  of  voltage  resonance  similar  to  that 
assumed  in  some  of  the  problems  of  Chapter  VI,*  and'  an  ex- 
cessive voltage  would  appear  between  A2  and  02  and  also  per- 
haps elsewhere  between  the  line  wire  connected  with  A2  and  the 
other  line  wires.  Evidently  with  the  conditions,  as  illustrated, 
the  primary  voltages  will  also  be  disturbed. 

Various  other  conditions  which  produce  objectionable  results 
may  arise  when  making  and  breaking  connections  of  trans- 
* Art.  81  (Examples  g to  k)  and  Art.  82  ; also  Art.  86. 


5G0 


ALTERNATING  CURRENTS 


formers  associated  with  polyphase  circuits,  so  that  circumspec- 
tion must  be  exercised  under  such  circumstances. 

The  balance  of  the  load  is  a most  important  element  in  using 
transformers  with  their  secondary  windings  connected  on  single- 
phase three-wire  or  on  polyphase  circuits.  It  may  be  readily 
understood  that  with  the  two  halves  of  the  secondary  winding 
of  one  transformer  connected  to  a single-phase  three-wire  system, 
any  unbalancing  of  the  load  will  affect  the  balance  of  the  second- 
ary voltages  respectively  produced  by  the  two  half  windings, 
on  account  of  the  fact  that  greater  magnetic  leakage  will  affect 
the  more  heavily  loaded  half  winding,  unless  the  two  halves 
are  very  closely  associated  on  the  magnetic  circuit.  It  is  there- 
fore important  to  have  each  transformer,  which  is  to  be  used 
for  this  purpose,  constructed  with  the  halves  of  its  secondary 
winding  so  arranged  that  the  magnetic  leakage  shall  be  ap- 
proximately the  same  in  each  even  if  the  load  becomes  unbal- 
anced. When  a core  t}Tpe  transformer,  with  the  primary  and 
secondary  windings  each  equally  divided  between  the  two  legs 
of  the  core,  is  to  be  used  to  supply  current  to  a three-wire  sec- 
ondary circuit,  the  half  of  the  windings  to  be  connected  be- 
tween either  outside  wire  and  the  neutral  wire  should  not  be 
placed  on  one  leg  of  the  core  only,  but  should  be  composed  in 
equal  parts  of  turns  on  each  of  the  legs.  When  transformers 
are  connected  in  polyphase  circuits  a similar  effect  of  unbalanced 
magnetic  leakage  is  produced  by  an  unbalanced  load,  as  may  be 
seen  by  reviewing  the  examples  given  earlier  in  the  book.* 
The  effect  of  such  unbalanced  loads  is  to  tend  to  overheat  the 
transformer  coils,  generating  apparatus,  and  load  appliances 
on  the  phases  carrying  abnormal  currents,  and  to  cause  unsat- 
isfactory regulation  and  danger  to  the  insulation  over  the  phases 
or  circuits  which  have  abnormal  voltages.  Not  only  do  incan- 
descent lamps  suffer  from  the  abnormal  voltages,  but  the  trans- 
former coils,  induction  motor  fields,  and  similar  apparatus  in  the 
circuits  having  excessive  voltages  are  apt  to  have  their  core  fluxes 
go  beyond  the  point  of  saturation,  thus  calling  for  excessive 
exciting  currents  which  may  become,  under  such  circumstances, 
sufficiently  largo  to  cause  serious  overheating.  Many  low 
voltage  secondary  distributing  systems  supply  three- wire  loads 
from  the  secondaries  of  transformers  connected  to  polyphase 


* Art.  103. 


MUTUAL  INDUCTION,  TRANSFORMERS 


561 


primaries,  and  there  have  been  frequent  cases  where  excellent 
plants  of  this  kind  have  given  most  unsatisfactory  service 
because  the  systems  have  not  been  properly  balanced.  Thus,  the 
three-wire  secondaries  of  the  transformers  may  be  unbalanced, 
and  in  addition  to  this  defect  the  sum  of  the  loads  of  the  trans- 
formers attached  to  one  phase  of  the  primary  supply  system 
may  be  very  different  from  the  loads  called  for  on  the  other 
phases.  The  line  and  generator  IR  drops  will  then  vary  in  the 
different  phase  circuits,  causing  the  vector  diagram  of  poly- 
phase voltages  to  become  skewed  and  the  voltages  of  the 
several  phases  at  the  points  of  power  consumption  to  become 
unequal.  It  is,  therefore,  essential  for  good  service,  when  con- 
necting transformers  to  compound  primary  or  secondary  circuits, 
to  first  properly  balance  the  load  which  is  to  be  attached. 

The  connections  of  quarter  and  tri-phase  transformers  are 
made  circuit  by  circuit,  in  the  same  manner  as  earlier  explained 
for  single-phase  transformers. 

141.  Phase  Transformation.  — Connections  for  transforming 
one  polyphase  system  into  another  system  with  a different 
number  of  phases  may  be  readdy  developed  from  the  principles 
set  forth  in  earlier  chapters. 

Quite  a number  of  commercial 
devices  for  this  purpose  have 
been  proposed.  The  one  most 
commonly  used  is  a T connec- 
tion arranged  as  follows  : In 
Fig.  317,  the  primary  wind- 
ings of  the  transformers  M and 
M'  are  alike  and  are  connected 
to  a quarter-phase  circuit. 

The  secondary  winding  OA^  of  M (sometimes  called  the  Teaser 
winding)  is  attached  to  the  middle  of  the  secondary  winding  of  M1 

at  the  point  0.  The  secondary  winding  of  M has  —B  times  the 


OR  LOAD 


Fig.  317.  ■ — Diagram  of  Arrangement  of  Two 
Transformers  for  converting  Quarter- 
phase  into  Tri-phase  System  and  Vice 
Versa.  The  Tee  Connection. 


number  of  turns  that  there  are  in  the  secondary  winding  of  M' . 
Then,  the  voltages  impressed  on  the  primary  windings  being 
numerically  equal  and  90°  apart  in  phase,  the  line  OB  in  Fig. 
314  represents  the  voltage  between  the  points  0 and  B2  in  Fig. 
317,  00  that  between  0 and  Cv  and  OA  that  between  0 and  A2. 
Voltage  OA  must  be  at  right  angles  to  OB  and  00 , as  the 


562 


ALTERNATING  CURRENTS 


THREE-PHASE  A 


fCl  j 

LESi 

M/WWWWv 

wwwwaaaT 

~lwvWW\AM 

phases  of  the  two  primary  circuits  are  90°  apart.  Thus,  the 
voltages  in  i?20ancl  OC2  being  in  phase  with  others  and  the 
voltage  in  OA2  being  in  quadrature  with  them,  it  is  seen  that 
between  the  points  A,  B , and  C three  equal  voltages  are  set 
up  120°  apart,  since  the  number  of  inducing  turns  between  0 
and  A2  bears  the  same  relation  to  the  number  of  inducing  turns 
between  B2  and  C2,  that  the  length  of  the  bisector  of  an  equi- 
lateral triangle  bears  to  one  of  its  sides.  The  apparatus  may 
be  used  with  equal  facility  for  transforming  two  phases  into 
three  phases  or  vice  versa,  by  impressing  two-phase  voltages  on 
AXPX  and  B101  and  thereby  producing  three-phase  voltages 
between  A2,  B2  and  C2  in  the  one  case,  or  by  impressing  three- 
phase  voltages  on  A2B2,  B2C2,  and  C2A2  and  thereby  producing 
quarter-phase  voltages  respectively  in  AXPX  and  BXCV 

For  some  services  it  is  desirable  to  use  a six-phase  system,  and 
to  do  this  economically  generally  requires  transformation  from 
quarter-phase  or  tri-phase  transmission  lines.  Figure  318  shows 
method  whereby  this  can  be  done  in  a simple  manner.  Here 

the  secondary  circuits  of  the  three 
transformers  of  a tri-phase  primary 
system  are  connected  at  opposite 
junctions  of  the  six  junction  points 
R , M,  S,  2F,  T,  P (Fig.  318)  of 
a mesh-connected  six-phase  load. 
The  load  makes  what  is  equivalent 
to  the  combination  of  two  tri-phase 
systems.  The  dotted  lines  between 
the  load  junctions  R,  31,  S,  W,  T, 
and  P represent  the  paths  of  the 
current  in  a mesh  load  and  the 
broken  diametral  lines  represent 
the  paths  for  a star  load.  In  six- 
phase  systems  mesh  and  star  connections  call  for  the  same  vol- 
tages and  currents  in  the  individual  branches  of  the  load. 
With  the  arrangement  shown  in  Fig.  318  the  neutral  point  of 
the  six-phase  star  is  at  the  same  potential  as  the  middle  point  of 
the  secondary  winding  of  each  of  the  three  transformers,  and  if 
a neutral  return  conductor  is  added  to  the  system,  it  may  be 
led  to  those  three  middle  points. 

In  Fig.  319  each  secondary  winding  is  shown  divided  into 


aaaaaa 


MA/W\ 


R 

LOAD 

P 


M/WV\ 


o 

:u 

N 


Fig.  318.  — Transformer  Connections 
for  transforming  Three  Phases  to 
Six  Phases.  Opposition  Arrange- 
ment. 


MUTUAL  INDUCTION,  TRANSFORMERS 


563 


two  equal  coils.  Each  set  of 
halves  is  connected  in  delta  ar- 
rangement, one  delta  being  re- 
versed compared  to  the  other  in 
accordance  with  the  connecting 
diagrams  shown  in  Fig.  319  a. 
The  corners  of  one  delta  are  con- 
nected in  tri-phase  mesh  to  three 
points  M,  N,  and  P on  the  load, 
while  the  other  delta  is  connected 
to  the  points  P,  S,  and  T.  This 
arrangement  is  commonly  called 
the  Double  delta  connection. 


Cl 


MAMAWvJwA WWmT 


B, 


Ai 

maammmJ 


Fig.  319.  — Transformation  of 
Three  Phases  to  Six  Phases  by  the 
Double  Delta  Arrangement. 


Figure  320  shows  a similar  arrangement,  but  with  the  two 


Fig.  319  a.  — Connection  Diagram  for  Plan  of  Fig.  319. 


Ci 


B. 


wmmvaaJ 


Ai 


sets  of  half  coils  connected  in  wye.  The  connection  diagram 

is  shown  in  Fig.  320  a. 

The  primary  windings  of  the 
transformers  may  be  in  either  delta 
or  wye  in  each  of  the  arrangements 
illustrated  for  the  transformation 
from  three  phases  to  six  phases  and 
the  reverse.  Which  arrangement 
to  select  for  either  the  primary 
windings  or  secondary  windings 
depends  upon  the  voltages  desired 
and  the  problems  of  insulation  and 
current-carrying  capacities  of  the 
copper  in  the  transformers,  second- 


A/VWY  4vVWy5lA/VVWL_, 

A/ywy'  3A/wvy'  s/wwy ' 


'!S 


''  N 


T 


Fig.  320.  — Transformation  of 
Three  Phases  to  Six  Phases  by  the 
Double  Wye  Arrangement. 


564 


ALTERNATING  CURRENTS 


ary  leads,  and  load.  With  a delta  system  the  line  current 
equals  V8  times  the  coil  current  for  the  balanced  three-phase 
system,  with  voltages  in  lines  and  coils  equal;  and  with  a wye 

M 


connection  the  line  voltages  equal  V3  times  the  coil  voltages, 
with  the  currents  in  lines  and  coils  equal.*  For  the  balanced 
six-phases,  line  currents  are  equal  to  coil  currents,  and  line 

voltages  are  equal  to  coil  vol- 
tages in  both  mesh  and  star, 
provided  line  voltages  are 
measured  between  electrically 
adjacent  terminals.  Figure 
320  b illustrates  the  vector 
relations  of  the  line  currents 
and  the  coil  currents  for  the 
six-phase  mesh.  The  line  cur- 
rent at  corner  M is  equal  to 
the  vector  sum  of  current  RM 
and  current  SM  (=  — MS) . 
This  is  the  current  Mb.  The 
line  current  at  corner  S is 
equal  to  the  vector  sum  of 
currents  MS  and  JVS,  and  so 
on  at  the  other  corners. 

Figure  321  shows  the  tee  method  of  connecting  two  trans- 
formers with  divided  secondary  windings  for  the  purpose  of 
transforming  from  three  to  six  phases.  In  this  case  the 
primary  winding  may  be  connected  to  a quarter-phase  circuit, 

* Art.  100. 


Fi«.  320  6.  — Vector  Diagram  of  Currents 
in  Six-phase  Mesh. 


MUTUAL  INDUCTION,  TRANSFORMERS 


505 


Ci 

MWiwlvWVWVW 


_/VW 


VwWNAA 


Fig.  321.  — Transformation  of  Two 
or  Three  Phases  to  Six  Phases 
using  Two  Transformers  and  the 
Tee  Connection. 


in  which  case  the  primary  windings  of  the  two  transformers 
must  be  alike  and  the  secondary  winding  of  the  teaser  trans- 
former must  contain  only  .866  ^ = as  many  turns  as  the  other 

secondary  winding,  or  to  a three-phase  primary  circuit,  in  which 
case  both  the  primary  and  secondary 
winding  of  the  teaser  transformer 
must  contain  VB/2  times  as  many 
turns  as  the  corresponding  winding 
of  the  other  transformer,  as  shown 
in  Fig.  321.  Other  combinations 
may  be  used,  but  those  given  are 
sufficient  to  indicate  in  general  the 
manner  in  which  the  transformation 
is  accomplished. 

Transformation  may  also  be  ef- 
fected from  a polyphase  circuit  to  a 
single-phase  circuit  for  the  purpose 
of  distributing  the  load  of  a single- 
phase circuit  over  the  several  phases  of  the  polyphase  circuit. 
Each  phase  of  a polyphase  circuit  is  a single-phase  circuit,  but 
when  it  is  desired  to  distribute  the  load  of  a single  single-phase 
circuit  over  the  three  phases  of  a three-phase  circuit  this  may 

be  done  by  connecting  three  trans- 
formers, as  illustrated  in  Fig.  321  a , 
which  also  exhibits  the  vector  diagram 
of  secondary  voltages.  It  will  be 
observed  that  the  connection  of  the 
secondary  winding  of  one  of  the 
transformers  is  reversed  in  compari- 
son with  the  normal  connection  for 
a delta,  and  the  connecting  wire  re- 
quired to  close  the  delta  is  omitted. 
The  vector  diagram  shows  that  the 
secondary  voltage  is  equal  to  twice 
that  of  one  transformer.  The  current 
carrying  capacit}r  is  equal  to  any  one 

of  the  transformers,  and  three 

transformers  are  required  to 
Single-phase  Load  over  Three  Phases.  transform  th.6  full  load,  of  two. 


Cl 

B, 

Ai 

Vwwww 

wwwwv 

VWAAAW 

56G 


ALTERNATING  CURRENTS 


Moreover,  the  results  are  not  to  be  recommended  because  the 
arrangement  transmits  the  pulsating  power  of  the  single-phase 
circuit  into  the  three-phase  circuit  and  unbalances  it. 

To  accomplish  the  corresponding  result  with  a two-phase  pri- 
mary circuit,  the  secondary  windings  of  the  transformers  should 
be  connected  as  for  use  with  a common  return  wire,  and  the 
single-phase  load  should  be  worked  off  the  outside  wires. 

The  transformation  of  single-phase  into  polyphase  currents 
by  means  of  stationary  transformers  may  be  accomplished  by 
phase-splitting  devices,  such  as  by  dividing  the  single-phase 
circuit  into  two  or  more  branches  and  obtaining  currents  of  the 
desired  polyphase  angles  by  inserting  property  proportioned 
resistances  and  inductive  and  capacity  reactances  into  the 
branches.  No  satisfactory  commercial  method  has  been  devel- 
oped, where  large  units  of  power  are  required,  which  does  not 
include  moving  parts  in  the  transformer,  though  phase-splitting 
devices  such  as  described  are  much  used  for  single-phase  inte- 
grating meters  and  for  starting  small  induction  motors. 

142.  Transformation  from  Constant  Voltage  to  Constant  Cur- 
rent. — - The  effect  of  magnetic  leakage  has  been  shown  in  Art. 
123.  Remembering  that  the  effect  is  like  that  produced  by  self- 
inductance coils  placed  in  the  primary  and  secondary  circuits,  the 
final  vector  diagram  is  the  same  as  in  the  case  of  a transformer 
working  on  an  inductive  secondary  circuit,  with  an  additional 
correction  applied  to  the  angle  of  lag  between  the  primary 
voltage  and  current  to  account  for  the  direct  effect  of  the  leak- 
age on  the  primary  circuit.  Such  a diagram  shows  that,  as  the 
leakage  is  increased  so  that  the  angle  of  lag  between  the  mutu- 
ally induced  voltages  and  the  secondary  current  approaches 
90°,  the  deficiency  in  the  inherent  tendency  to  regulate  for  con- 
stant secondary  voltage  when  constant  primary  voltage  is  im- 
pressed, becomes  so  great  that  the  secondary  terminal  voltage 
tends  to  vary  inversely  with  the  current  ; that  is,  the  voltage 
ordinates  of  the  transformer  regulation  curve  decrease  with  in- 
creasing rapidity  as  the  current  increases.  Such  a transformer 
should  therefore  tend  to  transform  a variable  current  at  con- 
stant voltage  into  a constant  current  at  a variable  voltage. 
This  characteristic  makes  transformers  with  large  magnetic 
leakage  desirable  for  use  for  providing  constant  current  for 
series  arc  lighting  by  transformation  from  a constant  voltage 


MUTUAL  INDUCTION,  TRANSFORMERS 


567 


circuit.  A transformer  constructed  for  use  in  this  way  is 
called  a Constant  current  transformer. 

When  the  lag  angle  between  secondary  voltage  and  current 
becomes  90°,  the  transformer  can  of  course  do  no  work,  conse- 
quently it  is  impossible  to  get 
very  exact  regulation  in  thus 
transforming  from  constant 
voltage  to  constant  current, 
but  it  is  possible  to  arrange 
the  transformer  so  that  the 
percentage  of  leakage  react- 
ance can  be  varied  when  nec- 
essary by  partially  closing  a 
shunt  magnetic  circuit  by  a 


[G.  322.  — Diagram  showing  a Simple  Way 
in  which  Magnetic  Leakage  can  be  Varied 
in  a Transformer  Core. 


proposed  by  Elihu  Thomson 
(Fig.  322). 

Figures  323  and  324  show  the  results  of  a test  of  a small 
transformer,  in  which  the  constant  current  regulation  is  wholly 
due  to  fixed  magnetic  leakage.  In  the  first  figure,  one  curve 

shows  the  efficiency 
as  a function  of  the 
current  in  the  sec- 
ondary circuit,  and 
the  other  curve  shows 
the  secondary  termi- 
nal volts  as  a func- 
tion of  the  secondary 
current.  The  sec- 
ond figure  has  curves 
A and  B which  show 
for  the  same  trans- 
formers the  watts  in 
the  primary  and  sec- 
ondary circuits  as 
functions  of  the  sec- 
ondary current. 
The  crosses  on  the  curves  in  the  two  figures  show  the  points 
corresponding  to  normal  load.  Close  regulation  in  the  trans- 
formation into  constant  secondary  current  from  constant  primary 


ppfUClENCY 



It* 

\ 

\ 

\ 

0 2 4 6 s 10  12 

AMPERES 

Fig.  323.  — Carves  of  Efficiency  and  Regulation  as 
functions  of  the  Secondary  Current  in  a Small  Trans- 
former having  High  Magnetic  Leakage. 


568 


ALTERNATING  CURRENTS 


voltage  requires  the 
use  of  accessory 
means,  such  as  means 
to  vary  the  magnetic 
leakage. 

In  well-built  trans- 
formers designed  for 
constant  secondary 
voltage,  magnetic 
leakage  is  not  likely 
to  be  of  much  mag- 
nitude, and  in  fact  it 
can  only  be  brought 
to  a large  value  by 
making  the  space  oc- 
cupied by  the  pri- 
mary and  secondary 
coils  very  large  com- 
pared with  the  cross  section  of  the  iron  core,  by  using  iron  of  a 
low  permeability,  or  by  specially  arranging  leakage  paths.  In 
Fig.  325  is  shown  a sketch  of  a modern  constant  current  trans- 
former in  which  it  may  be 
observed  that  the  magnetic 
circuit  is  long  and  the  space 
within  which  the  coils  are 
wound  is  large.  The  letters 
(7,  (Vindicate  the  laminated 
iron  core,  the  central  limb 
of  which  is  embraced  by  the 
primary  and  secondary  coils 
A and  B.  The  coil  A is 
movable  and  its  position  is 
determined  by  the  balance 
between  the  weight  of  the 
counter-weighted  coil  and 
the  magnetic  repulsion  be- 
tween the  coils.  Thus,  if 
the  repulsion  just  balances  the  weight  of  the  movable  coil 
and  mechanism  when  certain  currents  flow  in  the  two  coils, 
any  change  in  the  currents  will  cause  the  movable  coil  to 


Fig.  325.  — Constant  Current  Transformer. 


0 2 4 6 8 10  12 


AMPERES 

Fig.  324.  — Curves  of  Power  Input  and  Output  as  func- 
tions of  the  Secondary  Current  in  a Transformer 
having  High  Magnetic  Leakage.  A,  Primary  Kilo- 
watts ; B,  Secondary  Kilowatts. 


MUTUAL  INDUCTION,  TRANSFORMERS 


569 


move  up  or  down.  Instead  of  only  one,  both  coils  may  be 
made  to  move. 

The  transformer  being  connected  to  a secondary  circuit  com- 
prising translating  devices,  such  as  arc  lamps  in  series,  if  one 
of  the  units  of  load  is  short-circuited  for  the  purpose  of  remov- 
ing it  from  operation,  the  current  in  the  circuit  will  rise  on 
account  of  the  reduced  impedance.  Thereupon,  the  repulsion 
between  the  two  coils  is  increased  and  the  movable  coil  retreats 
farther  from  the  stationary  coil,  thereby  causing  an  increase  of 
the  magnetic  leakage  and  a reduction  of  the  secondary  terminal 
voltage.  The  counter-weight  being  suspended  on  an  arc  of 
slightly  decreasing  radius,  the  coil  promptly  finds  a position  of 
balance,  with  the  secondary  current  at  substantially  its  normal 
value.  If,  on  the  other  hand,  additional  load  is  added  to  the 
circuit,  the  current  will  at  once  correspondingly  decrease,  the 
repulsion  between  coils  also  decreases,  the  movable  coil  will 
move  toward  the  fixed  coil,  the  magnetic  leakage  will  be  thereby 
decreased  and  the  secondary  terminal  voltage  increased,  and  the 
coil  will  find  a new  position  of  balance  with  the  secondary  cur- 
rent again  at  substantially  its  normal  value. 

The  full  load  secondary  voltage  evidently  occurs  when 
the  coils  are  nearly  as  close  together  as  the  construction  per- 
mits, and  the  load  impedance  is  at  the  highest  value  through 
which  the  transformer  is  designed  to  send  the  normal  secondary 
current;  while  at  no  load,  which  is  the  condition  occurring 
when  the  secondary  circuit  is  short-circuited,  the  coils  are  at 
the  extreme  ends  of  the  central  magnetic  core.  When  a heavy 
load  is  suddenly  reduced,  preliminary  hand  regulation  may  be 
necessary  with  such  a transformer  to  prevent  unsafe  currents  or 
voltages  occurring  while  the  movable  coil  is  moving  to  its  proper 
position,  but  as  a rule  the  devices  are  considered  automatic.  The 
secondary  terminal  voltage  when  the  secondary  circuit  is  open, 
that  is,  when  the  load  impedance  is  infinite,  should  approach  a 
value  equal  to  the  primary  voltage  divided  by  the  ratio  of  trans- 
formation, as  in  such  a case  the  coils  will  be  at  their  nearest 
approach  to  each  other,  since  the  secondary  current  is  zero  and 
no  repulsion  exists  between  the  coils. 

The  circle  diagram  for  such  a transformer  when  the  load  is 
considered  to  have  a power  factor  of  unity  and  the  ratio  of 
transformation  is  unity  is  shown  in  Fig.  326.  For  con- 


570 


ALTERNATING  CURRENTS 


venience  in  this  figure  the  negative  resistance  loci  are  used  for 
representing  secondary  quantities.  In  the  case  under  consider- 
ation the  reactance  and  resistance  of  the  equivalent  circuit  both 
vary  and  in  such  proportion  that  the  secondary  current  is  main- 
tained constant.  The  secondary  current  locus  is  therefore  rep- 
resented by  the  semicircle  A,  I2",  I2,  I2,  B.*  The  construc- 
tion is  otherwise  of  the  same  principles  as  shown  in  Figs.  277 
to  281  inclusive.  At  full  load  the  leakage  reactance  voltage 
drop,  represented  by  the  line  Q1BV  is  at  the  relatively  low  value 

which  occurs  when  the  trans- 
former coils  are  close  together. 
At  no  load,  that  is,  the  secondary 
short-circuited,  the  leakage  re- 
actance voltage  drop  and  the 
resistance  drop  in  the  two  coils 
must  combine  to  equal  the  im- 
pressed voltage  OEv  The  leak- 
age voltage  then  has  the  value 
of  EXQ".  The  resistance  drop 
remains  numerically  equal  to 
K^QV  since  the  secondary  cur- 
rent is  maintained  constant  and 
the  ampere-turns  of  the  primary 
winding  therefore  must  stay  con- 
stant, except  as  the  variation  of 
the  magnetic  leakage  may  affect 
the  losses  and  thus  affect  the 
exciting  component  of  the  cur- 
rent. That  is,  reducing  the 
series  load  by  short-circuiting 
the  translating  devices  does  not 
decrease  the  primary  current,  but  it  causes  the  primary  angle 
of  lag  to  increase  and  therefore  reduces  the  power  absorbed 
from  the  primary  supply  circuit.  The  primary  current  is  found 
as  in  the  constructions  in  the  earlier  figures  referred  to,  and  is 
shown  for  one  position  of  Q by  the  line  SXIV  where  S10  is  the 
exciting  component. 

If  the  load  impedance  is  increased  beyond  the  point  required 
to  bring  the  two  coils  of  the  transformer  to  their  position  of 
* Compare  with  Art.  70. 


E, 


Fig.  326.  — Diagram  showing  Circular 
Loci  for  Constant  Current  Trans- 
former. 


MUTUAL  INDUCTION,  TRANSFORMED 


571 


nearest  approach,  which  represents  the  condition  of  maximum 
load,  the  primary  and  secondary  currents  will  decrease  along  a 
locus  determined  by  the  effects  of  a varying  resistance^  and 
constant  reactance  in  a circuit.* 

The  equivalent  impedance  combination  required  to  represent 
the  reactions  of  this  type  of  transformer  can  be  built  up  as  in  the 
case  of  constant  voltage  transformers  for  any  value  of  the  load, 
but  the  equivalent  coil  reactance  is  not  fixed  but  varies  with  the 
position  of  the  movable  coil,  and  the  parallel  impedance  sup- 
plying the  exciting  current  should  be  connected  at  the  position 
indicated  by  JYS  in  Fig.  282,  since  the  mutually  induced  vol- 
tage varies  widely. 

Figure  329  shows  a constant  current  transformer  at  A,  which 
has  a double  secondary  circuit  arranged  to  operate  two  series  of 
arc  lamps.  A plug  switch  is  provided  in  each  secondary  ex- 
ternal circuit,  as  shown  at  £,  to  be  used  for  short-circuiting 
either  one  at  will.  Other  plug  switches  are  provided  at  C and 
D to  be  used  for  disconnecting  the  external  secondary  circuits 
from  the  transformer  and  for  disconnecting  the  transformer 
from  the  primary  supply  circuit.  An  amperemeter  is  arranged 
so  that  it  may  be  introduced  into  either  lighting  circuit  by 
means  of  a switchboard  plug,  and  a lightning  arrester  is  con- 
nected to  each  circuit. 

143.  Series  Transformers.  — When  transformers  are  connected 
in  series  with  a line  carrying  a load  as  shown  in  Fig.  327  they  are 
usually  called  Series  or  Current  transformers.  Such  transformers 
in  commercial  service  are  usually  of  very  small  capacity,  such  as 
for  loads  of  from  ten  to  fifty  watts  in  the  secondary  circuit,  and 
are  provided  for  service  with  electrical  measuring  instruments, 
circuit  breaker  relays,  and  the  like.  The  current  in  the  second- 
ary coil  divided  by  the  ratio  of  transformation  takes  a value 
equal  to  the  vector  difference  between  the  total  primary  current 
and  the  exciting  current  component.  The  exciting  current 
varies  with  the  load  for  the  reasons  next  explained  ; but  in  a well- 
designed  machine  it  is  relatively  small  for  all  conditions  of  load 
from  a short  circuit  of  the  secondary  terminals  up  to  an  imped- 
ance in  the  secondary  circuit  which  is  considerably  greater  than 
that  of  normal  full  load.  The  voltage  across  the  primary 
terminals  equals  the  whole  transformer  impedance,  including 

* Art.  70. 


572 


ALTERNATING  CURRENTS 


that  of  the  secondary  load,  reduced  to  primary  equivalents,  multi- 
plied by  the  primary  current ; consequently,  when  the  primary 
current  is  constant,  if  the  impedance  of  the  load  increases,  the  sec- 
ondary current  tends  to  fall, but  the  exciting  component  increases 
so  as  to  maintain  the  closed  vector  relation  between  secondary 
ampere-turns,  exciting  ampere-turns,  and  total  primary  ampere- 
turns.  If  the  impedance  of  the  load  decreases,  the  secondary  cur- 
rent tends  to  increase  and  the  exciting  component  decreases. 
The  tendency,  therefore,  is  for  the  magnetic  flux,  and  therefore 
the  secondary  voltage,  to  vary  so  as  to  maintain  the  secondary 
current  in  approximately  fixed  relation  to  the  primary  current. 

The  secondary  mutually  induced  voltage  equals  the  primary 


< GENERATOR  LOAD 

WW 


Fig.  327. — Connections  for  a Current  Transformer. 

mutually  induced  voltage  divided  by  the  ratio  of  transformation. 
For  the  conditions  of  constant  impedance  in  the  secondary  cir- 
cuit, the  secondary  current  is  therefore  substantially  proportional 
to  the  primary  current  and  in  nearly  opposite  phase,  so  long  as 
the  maximum  magnetic  density  in  the  transformer  core  is  suffi- 
ciently well  below  the  point  of  saturation  so  that  it  may  be 
considered  to  be  substantially  proportional  to  the  exciting 
current.  When  the  transformer  secondary  load  is  the  current 
coil  of  a wattmeter  or  an  amperemeter,  as  is  usually  the  case, 
this  condition  is  approximately  realized. 

If  the  load  impedance  is  increased  so  that  the  saturation  of 
the  magnetic  core  is  high,  a large  portion  of  the  primary  current 
may  be  used  for  exciting  purposes,  leaving  a comparatively  small 
portion  equal  and  opposite  to  the  equivalent  secondary  current. 
It  is  thus  seen  that  the  ratio  of  the  secondary  current  to  the 
primary  current  may  vary  considerably  in  case  the  impedance  of 
the  secondary  circuit  varies,  especially  if  the  latter  is  rather  large. 


MUTUAL  INDUCTION,  TRANSFORMERS 


573 


so  that  when  a series  transformer  is  used  to  supply  current  to  the 
coils  of  a measuring  instrument,  it  should  be  used  only  with  in- 
struments with  which  it  was  designed  to  be  associated,  or  the 
instruments  should  be  calibrated  in  association  with  the  trans- 
former. When  more  than  one  instrument  coil  (such  as  the 
current  coil  of  a wattmeter  and  the  coil  of  an  amperemeter)  is 
to  be  associated  with  a particular  series  transformer,  the  in- 
strument coils  should  be  connected  in  series  with  each  other  in 
the  transformer  secondary  circuit.  The  transformer  should  be 
designed  so  that  the  magnetic  density  in  the  core  is  sufficiently 
far  below  the  point  of  saturation  so  that  the  exciting  current  is 
small  and  approximately  proportional  to  the  impressed  voltage 
and  hence  to  the  secondary  current,  over  the  entire  range  of  the 
readings  of  the  instruments  from  zero  to  their  maximum 
readings. 

When  the  secondary  circuit  of  such  a transformer  is  opened, 
the  total  current  flowing  through  the  mains  into  which  the 
primary  coil  is  inserted,  acts  as  an  exciting  current.  The  form 
of  the  wave  of  this  current  is  largely  fixed  by  the  constants  of 
the  load  attached  to  the  mains,  a small  current  transformer 
having  little  effect  upon  it,  so  that  the  mutually  induced  vol- 
tage and  the  secondary  current  may  be  very  irregular  in  form  if 
the  magnetization  of  the  core  is  carried  beyond  the  point  of 
saturation.  This  is  the  same  condition  as  is  shown  in  Fig.  264, 
which  gives  the  curves  of  current,  magnetism,  and  induced 
voltage  in  an  induction  coil  through  which  a current  of  fixed 
form  flows.  * The  secondary  voltage  when  the  secondary  circuit 
is  open  depends  upon  the  magnetic  reluctance  of  the  core,  the 
current  in  the  main  wires,  and  the  ratio  of  the  windings.  As 
the  magnetic  density  in  current  transformers  designed  to  be 
used  for  measuring  instruments  is  made  very  low  through  the 
range  of  currents  to  be  measured,  it  is  evident  that  the  second- 
ary circuit  voltage  may  rise  to  a high  and  dangerous  value 
when  a large  current  flows  in  the  mains,  if  the  secondary  circuit 
is  open,  and  in  any  event  the  magnetic  flux  is  likely  to  rise 
to  an  excessive  density  and  cause  injurious  heating  from  the 
hysteresis  and  eddy  current  losses.  The  secondary  coils  of 
such  transformers  can  be  short-circuited  at  any  time,  however, 
without  danger,  as  they  will  not  then  take  a greater  current 


* Art.  116. 


574 


ALTERNATING  CURRENTS 


than  that  in  the  primary  circuit  divided  by  the  ratio  of  trans- 
formation. It  is  a safe  rule  never  to  open  the  secondary  circuit 
of  an  instrument  transformer  when  current  is 
flowing,  and  it  should  be  short-circuited  when 
connected  to  a supply  circuit  but  not  provided 
with  a load. 

Figure  328  represents  a commercial  current 
transformer  complete  in  a neat  iron  case. 
Figure  329  shows  a current  transformer  at 
F and  a potential  transformer  at  F connected 
to  a registering  watt-hour  meter. 

When  a wattmeter  is  used  in  association 
with  a current  transformer  and  a potential 
Fig  328.  — Current  transformer,  a good  deal  of  caution  should  be 
Transformer  in  its  observed  in  respect  to  its  readings  unless  the 
Case'  instrument  has  been  calibrated  in  association 

with  the  same  transformers  and  with  due  regard  to  the  power 
factor  of  the  load  which  it  is  to  be  used 
to  measure.  The  accuracy  of  the  watt- 
meter readings  is  dependent  upon 
the  preservation  of  the  same  angular 
relation  between  the  current  in  its 
current  coil  and  the  voltage  impressed 
on  its  voltage  coil  as  exists  between 
the  main  circuit  current  and  voltage  ; 
and  also  upon  the  transformation  of 
both  current  and  voltage  occurring  in 
fixed  ratios.  The  potential  trans- 
former, being  a small  constant  vol- 
tage transformer,  its  secondary  terminal 
voltage  differs  in  phase  very  nearly 
180°  from  the  phase  of  the  primary 
impressed  voltage,  but  there  is  a very 
small  deviation  from  an  exact  180°  re- 
lation which  is  caused  by  the  magnetic 
leakage  and  the  primary  IR  drop.  In 
the  current  transformer,  the  secondary  Fig_  329.  _ constant  Current 
current  differs  substantially  180°  from  Transformer  with  Double  See- 
the phase  of  the  primary  current  except 


t 

Tl 

buAh 


T TT-T.  I LIGHTNING 

JJG  13 ARRESTER 


iJlU 

C(y)  C($)  swishes  (§)C  (y)C 

MMETEI 

Q 


f 


* 


=7 


B 


CONSTANT 

CURRENT 

TRANSFORMER 


WATTHOUR 

METER  4/T 


=-i- 


RL 


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)(|) 


for  the  effect  of  the  exciting  current. 


ondary  Circuit  connected  to 
Series  Load.  Also  Instrument 
Transformers. 


MUTUAL  INDUCTION,  TRANSFORMERS 


575 


Each  of  these  causes  of  inaccuracy  in  apparent  lag  angle  is 
quite  small  in  good  commercial  transformers  and  has  relatively 
little  effect  on  readings  of  the  instrument  when  the  load  has  a 
high  power  factor,  but  even  such  small  deviations  may  intro- 
duce a large  error  into  the  readings  when  the  power  factor  of 
the  load  is  small. 

144.  Impedance  Coils,  Compensators,  etc.  — The  design  of 
Reactance  coils,  Impedance  coils,  or  Choking  coils,  as  they  are 
variously  called,  is  carried  out  in  very  much  the  same  manner 
as  the  design  of  a transformer.  An  impedance  coil  consists  of 
a magnetic  circuit  with  a winding  of  small  resistance  but  large 
inductance.  It  may  be  used  in  lieu  of  a rheostat  to  modify  the 
current  in  an  alternating-current  circuit  and  with  less  loss  of 
power  than  would  be  caused  by  the  rheostat.  The  magnetic 
circuit  and  winding  are  proportioned  in  exactly  the  same  manner 
as  the  magnetic  circuit  and  the  primary  winding  of  a trans- 

Coils  of  this  type 


former,  using  the  formula  Ex  = 


are  used  for  a variety  of  purposes  where  it  is  desired  to  throttle 
the  flow  of  current  without  the  attendant  loss  of  power  which 
always  follows  the  use  of  resistances. 

Thus  where  an  arc  lamp  is  used  on  a constant  voltage  alternat- 
ing-current circuit,  a reactance  coil  is  ordinarily  used  to  re- 
duce the  voltage  from  the  voltage  on  the  wires  to  that  required 
at  the  lamp  and  to  aid  in  maintaining  the  stability  of  the  flow 
of  current.  Such  an  arrangement  is  illustrated  in  Fig.  330, 
where  the  circuit 
connections  are  in- 
dicated at  the  left 
hand,  and  the  vector 
diagram  of  voltages 
measured  across  the 
line,  between  the  arc 
terminals,  and  be- 
tween the  coil  ter- 


Y 

REACTANCE  COIL 

(arc 

Fig.  330.  — Reactance  Coil  in  Series  with  an  Arc  Lamp. 


minals,  are  indicated  at  the  right-hand.  The  voltages  upon  the 
distributing  circuits  in  a theater,  or  upon  the  feeders  of  any 
plant  furnishing  alternating  currents  for  incandescent  lighting, 
may  be  regulated  by  impedance  coils.  Figure  331  shows  one 
of  the  original  Thomson  impedance  coils,  in  which  the  reactive 


576 


ALTERNATING  CURRENTS 


Fig.  331.  — An  Early  Type  of  Voltage  Regulator  in 
which  the  Reactance  of  the  Coil  is  varied-  by  Means 
of  a Movable  Copper  Shield. 


effect  is  varied  by  moving  a heavy  copper  shield  A so  as  to 
more  or  less  inclose  the  winding  B,  instead  of  varying  the 

number  of  turns  of  the 
winding  included  in  the 
circuit.  This  shield  acts 
like  the  short-circuited  sec- 
ondary of  a transformer, 
and  therefore  reduces  the 
apparent  impedance  of  the 
windings  as  it  approaches 
them.  Figure  332  is  a reg- 
ulator for  a constant-cur- 
rent circuit  such  as 
may  be  conveniently 
used  in  series  lighting 
circuits,  in  which  A 
is  the  impedance  coil  winding  and  B is  an  iron  core  for  the 
magnetic  circuit.  By  means  of  the  regulating  mechanism  (7, 
the  core  B is  automatically  thrust  varying  distances  into  A, 


thus  holding  the  load  current  con- 
stant very  much  as  is  done  in  a 
constant  current  regulating  trans- 
former.* 

It  is  evident  that  the  reactance 
of  an  impedance  coil  is  more  eco- 
nomical for  use  in  reducing  voltages 
than  a simple  resistance,  since  pure 
reactance  in  a circuit  neutralizes 
part  of  the  impressed  voltage  with- 
out absorbing  power,  though  it  has 
the  disadvantage  of  lowering  the 
power  factor. 

There  is  another  type  of  induc- 
tive apparatus  which  is  much 
used  and  which  goes  under 
the  name  of  Autotransformer  or 
Compensator.  This  consists  of  a single  winding  on  a proper 
magnetic  circuit,  to  which  single  winding  both  the  primary  and 
secondary  circuits  are  connected.  Figure  333  shows  the  connec- 


Fig.  332.  — Regulator  for  a 
Current  Circuit. 


Constant 


* Art.  142. 


MUTUAL  INDUCTION,  TRANSFORMERS 


577 


tions  of  a 220-volt  compensator  which  feeds  two  110-volt  sec- 
ondary circuits.  In  this  case  the  function  of  the  compensator 
is  to  equalize  the  voltage  between  the  two  secondary  circuits  re- 
gardless of  their  relative  loads.  This  purpose  is  fulfilled  fairly 
well,  although  when  the  two  loads  are  unbalanced  the  voltages  de- 
livered by  the  two  halves  of  the  compensator  are  different,  due 
to  the  differences  of  drop  caused  by  the  local  impedances  of  the 
two  halves  with  different  currents  flowing  through  them. 
When  the  load  is  c 
balanced,  no  current 
flows  through  the 
neutral  wire  which 
is  connected  to  the 
compensator  at  0 , 
and  the  compen- 
sator winding  car- 
ries  Only  sufficient  Fig.  333. — Compensator  used  for  Maintaining  the  Vol- 
current  to  excite  taSes  of  a Three-wire  Circuit. 

the  core  so  that  a counter-voltage  is  produced  which  is  equal 
and  opposite  to  the  vector  difference  of  the  impressed  voltage 
and  the  IR  drop  caused  by  the  exciting  current.  Upon  unbal- 
ancing the  load  more  power  is  absorbed  on  one  side  of  the 
neutral  wire  than  on  the  other,  and  a local  current  flows  through 
the  neutral  wire  and  part  of  the  compensator  winding.  In  this 
case  the  halves  of  the  compensator  act  as  a transformer.  Con- 
sidering the  case  when  one  side,  A,  of  the  three-wire  circuit  is 
fully  loaded,  and  the  other  side,  B,  is  without  load,  — then, 
neglecting  the  compensator  core  losses  and  exciting  current, 
the  current  in  wire  6r  is  zero  and  the  current  in  wires  E 
and  F is  twice  as  great  as  the  current  in  wires  Q and  D,  since 
the  power  delivered  all  comes  from  the  primary  circuit  but  the 
delivery  voltage  is  only  one  half  as  large  as  the  primary  voltage. 
The  compensator  must  operate  as  a transformer  to  transform 
one  half  of  the  power  required  for  the  load  from  the  B side  of  the 
circuit  to  the  A side  of  the  circuit.  The  current  in  the  neu- 
tral wire  therefore  divides  at  0 into  two  halves,  which  flow  re- 
spectively in  opposite  directions  through  the  winding.  A 
similar  condition  exists  for  any  degree  of  unbalance,  so  that 
one  half  the  current  in  the  neutral  wire  flows  in  opposite  direc- 
tions through  the  two  halves  of  the  compensator  winding  when  it 

2 p 


578 


ALTERNATING  CURRENTS 


is  used  to  halve  the  primary  voltage  as  in  this  instance,  and  the 
wire  of  the  winding  need  be  made  only  large  enough  to  carry 
without  overheating  one  half  the  current  in  the  neutral  wire  at 
the  time  of  the  maximum  degree  of  unbalance  that  can  occur  in 
the  circuit  concerned.  A compensator  therefore  requires  less 
copper  under  these  conditions  than  is  required  for  a regular 
transformer. 

Figure  334  shows  the  connections  of  a 330-volt  compensator 

which  supplies  a 1100-volt  secondary  circuit  or  vice  verm.  In 

this  arrangement  suppose 

that  a,  b are  the  terminals 

of  the  primary  circuit,  and  b, 

c of  the  secondary  circuit. 

Then  the  exciting  current  is 

an  element  of  the  primary 

current,  and  passes  from  a 

to  b.  The  induced  voltages 

between  a and  b,  and  b and 

c are  proportional  to  the  num- 

Fig.  334.  — Compensator  arranged  to  Trans-  her  of  turns  between  the  ter- 

form  Voltage  from  uoo  to  330  Volts  or  minals,  as  in  a transformer. 

Vice  Versa.  rr,,  , , • ,■ , 

lhe  secondary  current  is  the 

combination  of  two  currents,  flowing  in  ac  and  be,  which 
produce  equal  and  opposite  magneto-motive  forces.  Neglect- 
ing the  core  losses  and  exciting  current,  the  current  in  ac  is 
equal  to  the  current  in  the  secondary  circuit  divided  by  the 
ratio  of  the  turns  in  ab  to  the  turns  in  cb,  that  is,  divided  by 
the  ratio  of  transformation,  since  the  power  delivered  to  the 
secondary  circuit  all  comes  from  the  primary  circuit.  The 
current  in  the  windings  between  terminals  cb  must  therefore 
be  less  than  the  current  in  the  secondary  circuit  in  the  ratio  of 

turns  ab  — turns  cb  ___  turns  ab 
turns  cb  turns  cb 

The  current  in  the  winding  between  c and  b is  therefore  equal 
to  the  difference  between  the  secondary  current  and  the  primary 
current.  Thus,  it  is  evident  that  the  autotransformer  acts  very 
much  like  a transformer ; and  modified  transformer  diagrams  and 
arrangements  of  substituted  impedance  can  be  as  readily  worked 
out  for  them  as  for  transformers.  The  sizes  of  wire  in  ac  and 


MUTUAL  INDUCTION,  TRANSFORMERS 


579 


Fig.  335.  — Connection  of  Compensators  to  a Tri-phase 
Circuit  in  Wye,  — Step-down  or  Step-up. 


be  must  be  made  with  reference  to  the  current  in  each  part  at 
full  load,  and  the  two  parts  should  be  sandwiched  together  or 
otherwise  arranged 
so  as  to  keep  down 
magnetic  leakage  to 
the  limit  required. 

The  magnetic  circuit 
is  made  of  the  same 
materials  and  is  sub- 
ject to  the  same  con- 
ditions as  are  the 
cores  of  transform- 
ers, but  for  equal  cur- 
rent densities  in  the 
windings  the  weight 

of  copper  is  less  in  the  autotransformer  by  an  amount  proportional 

to  twice  the  weight  of 
the  primary  winding 
divided  by  the  ratio 
of  transformation. 

Autotransformers 
maybe  connected  in 
wye  to  a tri-phase 
circuit  as  shown  in 
Fig.  335,  when  the 
secondary  parts  are 
grouped  around  the 
neutral  point;  or  in 
delta,  as  shown  in 
Fig.  336,  though  in  this  case,  as  can  be  seen  from  the  figure, 
the  lowest  secondary 
voltage  possible  is 
one  half  of  the  pri- 
mary voltage. 

Figure  337  shows 
autotransformers  con- 
nected into  a tri-phase 
circuit  for  starting  a 
motor  at  lower  vol- 
tage than  that  of  the 


Fig.  336. — Connection  of  Compensators  to  a Tri-phase 
Circuit  in  Mesh — Step-down  or  Step-up. 


GENERATOR 


Fig. 


337.  — Compensators  connected  for  starting 
Motor  at  Reduced  Voltage. 


580 


ALTERNATING  CURRENTS 


line.  Here  any  one  of  three  sets  of  secondary  taps  may  be  used. 
The  switching  devices  are  so  arranged  that  the  autotransformer 
is  only  in  series  during  the  starting  period.  Polyphase  auto- 
transformers  can  be  built  with  the  cores  like  those  of  polyphase 
transformers.  Regulation  or  starting  taps  can  be  taken  from 
the  polyphase  secondaries  of  banks  of  ordinary  transformers  used 
in  supplying  load  to  motors  or  other  apparatus.  The  connec- 
tions can  in  such  cases  be  made  similar  in  principle  to  those 
shown  in  Fig.  337  or  sometimes  as  those  in  Figs.  335  and  336. 
Where  special  transformers  are  used  for  supplying  a motor  this 
arrangement  is  desirable,  as  the  transformers  thus  perform  the 
dual  functions  of  furnishing  the  proper  operating  voltage  to 
the  machine  and  acting  as  a regulator  or  starter. 

One  or  both  of  the  secondary  terminals  of  an  autotransformer 
may  be  arranged  so  that  the  position  of  connection  to  the 
winding  may  be  varied,  and  by  that  means  the  secondary 
voltage  may  be  caused  to  vary  through  any  desired  range, 
while  the  primary  current  changes  only  so  far  as  is  required  by 
any  change  in  the  power  absorbed  b}r  the  secondary  circuit. 
The  last  step  may  bring  the  secondary  circuit  terminals  into 
connection  with  the  primary  circuit  terminals,  thus  putting  full 
voltage  on  the  secondary  circuit,  as,  for  instance,  in  the  case  of 
autotransformers  in  three-phase  circuits,  the  final  step  puts  the 
secondary  terminals  in  connection  with  the  primary  terminals, 
as  shown  by  a,  5,  c of  Figs.  335  and  336.  Autotransformers 
are  commonly  used  for  voltage  regulators  in  intermittent  serv- 
ice, as,  for  instance,  in  place  of  rheostats  for  starting  alternat- 
ing current  motors.  They  are  evidently  lighter  in  weight  per 
kilowatt  of  energy  transmitted,  and  are  therefore  desirable  for 
use  when  neither  the  primary  nor  secondary  voltages  are  of 
such  a high  value  that  it  is  unwise  to  have  the  primary  and 
secondary  circuits  electrically  connected  together. 

Figure  338  shows  a small  autotransformer  intended  for  vol- 


tage regulation,  which  may 
he  used  either  to  reduce 
the  voltage  in  the  circuit 


Fig.  338.  — Auto  transformer  arranged  telescop- 
ically so  that  it  can  he  used  as  a Regulator. 


(in  which  case  it  is  called 
a Dimmer)  or  to  raise  it  (in 
which  case  it  is  called  a Booster)  depending  upon  how  its  internal 
connections  are  made.  The  regulation  is  effected  by  pushing 


MUTUAL  INDUCTION,  TRANSFORMERS 


581 


Fig.  339.  — Transformer  Windings  ar- 
ranged in  Shell  Transformer  of  Given 
Dimensions  so  that  Leakage  is  large. 


a part  of  the  winding  in  or  out  of  the  solenoid  formed  by  the  re- 
mainder. Numerous  devices  of  this  kind,  which  depend  upon 
moving  the  primary  and  secondary  coils  with  reference  to  each 
other,  or  the  core  with  reference  to  both,  are  manufactured. 

145.  Calculation  of  Magnetic  Leakage.  — By  properly  plac- 
ing the  windings  with  respect  to  each  other  and  to  the  magnetic 
circuit,  the  magnetic  leakage 
of  transformers  may  be  re- 
duced to  reasonably  small 
values.  Thus  in  Fig.  339  the 
primary  and  secondary  wind- 
ings P and  S are  so  placed 
that  a short-circuiting  of  mag- 
netic flux  along  the  path  indi- 
cated by  the  dotted  lines  is  to 
be  expected  ; but  when  the 
windings  are  arranged  as 
shown  in  Fig.  340,  the  leakage  is  not  as  great,  because  of  the 
greater  reluctance  in  the  magnetic  circuits  of  the  leakage  paths 
caused  by  their  greater  length  and  lesser  cross  section;  while  if 
each  of  the  windings  is  divided  and  the  primary  and  secondary 
parts  sandwiched  together,  the  leakage  maybe  made  very  small. 

The  magnetic  leakage  may  be 
calculated  with  approximate 
accuracy  by  the  method  indi- 
cated below. 

Since  the  currents  in  the 
primary  and  secondary  coils 
of  the  transformer  are  in  prac- 
tical opposition  of  phase,  their 
magnetizing  effects  are  oppo- 
site. This  tends  to  cause 
lines  of  force  to  short-circuit 
through  the  coils,  as  shown  in  Fig.  339,  the  tendency  being 
greatest  at  the  plane  where  the  coils  touch  each  other,  since 
the  magneto-motive  force  is  there  the  greatest,  and  falling  off 
to  zero  at  the  outer  edges  of  the  coils,  so  that  the  magnetic 
leakage  will  differ  for  each  layer  of  wire  in  the  coils.  The 
effect  of  leakage  must  therefore  be  calculated  for  each  layer, 
and  the  total  effect  may  then  be  summed  up. 


Fig.  340.  — Transformer  Windings  ar- 
ranged in  the  Shell  Transformer  shown  in 
Fig.  339,  but  so  that  Leakage  is  reduced. 


582 


ALTERNATING  CURRENTS 


C' 


In  Fig.  341  the  ordinates  of  the  line  A 'BA"  are  proportional 
to  the  total  ampere-turns  acting  at  any  point  to  cause  leakage 

lines  to  pass  through  the  coils  P 
and  S.  These  ordinates  are  equal 
to  the  number  of  turns  in  a coil 
between  the  foot  of  the  ordinate 
and  the  outer  edge  of  the  coil, 
multiplied  by  the  current  flowing 
in  the  turns.  The  ordinate  at  the 
outer  edge  of  each  of  the  windings 
is  evidently  zero,  and  at  the  plane 
between  the  windings  reaches  a 
maximum  equal  to  practically  n2I2. 
The  number  of  the  leakage  lines 
of  force  inclosed  by  any  layer  is 
proportional  to  the  corresponding 
when  these  ordinates  are  respec- 


A 

1 

1 

1 

1 

1 

1 

ff>  1 
*7  1 

\ B / 

1 

1 

^ • 
I 

1 

1 

1 

1 

__  ^ 

72'2,I2 

i 

1 

A' 

A" 

P s 

Fig.  341.  — Diagram  for  showing  the 
Relation  of  Transformer  Leakage 
Magneto-motive  force  to  Magnetic 
Leakage. 


ordinate  of  the  lines  C DC" 
tively  equal  to 

1.25V2  ya 


x = 


l 


where  x is  the  desired  ordinate,  y is  the  mean  ordinate  of  the 
line  A' BA"  taken  from  the  neutral  plane  at  D to  the  point 
under  consideration,  a = m n is  the  area  of  the  coil  between  the 
neutral  plane  and  the  point  under  consideration,  n being  the 
length  of  the  coil  perpendicular  to  the  plane  of  Fig.  342,  and  l 
is  the  average  length  of  the  leakage 
lines  of  force  through  the  coils 
(Fig.  342).  The  maximum  value 
of  x falls  at  the  outer  edges  of  the 
coils  and  is 

_1.25V2V,a, 

**  m 


21 


Fig.  342. — Diagram  showing  the 
Dimensions  of  Leakage  Paths  in 
a Transformer. 


where  Ax  is  the  total  iron  surface 
presented  to  a coil  from  which  leak- 
age lines  emerge ; and  the  average  number  of  leakage  lines 
inclosed  by  the  different  layers  at  the  instant  of  maximum 


leakage  is 


1.2->  d2I2A^  _ 

2V2  l 


,45  # 


MUTUAL  INDUCTION,  TRANSFORMERS 


583 


The  inductive  effect  of  this  leakage  on  the  secondary  winding 
is  equal  to 

/7T  mf 

V 2 7 rn2 

108  ; 


and  an  equivalent  effect  is  produced  on  the  secondary  voltage 
on  account  of  the  leakage  of  lines  of  force  through  the  primary 
coil.  If  A is  taken  to  represent  the  total  area  of  iron  presented 
to  both  windings  from  which  leakage  lines  emerge,  the  for- 
mulas become 


<J>  = l-25  n2I2A  d E = V2  t m^J  = ±n22I2Af 
1 V2 1 108  108Z  ’ 


where  is  the  combined  leakage  voltage  of  the  primary  and 
secondary  coils,  the  former  reduced  to  secondary  equivalents. 
The  inductive  effect  due  to  magnetic  leakage  lias  already  been 
shown  to  be  in  quadrature  with  the  mutually  induced  voltages  of 
the  transformer.*  The  effect  of  the  leakage  voltage  is  deter- 
mined by  the  formulas  or  diagrams  already  given,  f The  formu- 
las given  above  are  for  the  parts  of  the  coils  under  the  iron  only, 
and  the  iron  portion  of  the  leakage  paths  was  assumed  to  be  of 
negligible  reluctance.  Leakage  around  the  ends  of  the  coils 
has  some  effect,  and  the  leakage  paths  in  the  iron  have  a low 
reluctance  which  cannot  always  be  considered  negligible.  As 
a result  the  formulas  derived  do  not  give  the  total  magnetic 
leakage  exactly  and  though  approximately  correct  they  should 
be  used  with  discretion. 

146.  Current  Rushes  and  Surges.  — It  was  shown  earlier  that 
the  exponential  term  in  the  complete  equation  for  an  alternating 
current  in  an  inductive  circuit  is  ordinarily  negligible,  but 
under  certain  conditions  its  effect  for  a few  periods  after  the  cur- 
rent is  started  in  a circuit  may  be  considerable.  This  question 
was  investigated  by  Fleming  | and  others  § with  especial  refer- 
ence to  the  action  of  transformers  when  first  switched  on  to  an  al- 
ternating current  circuit.  If  a transformer  is  switched  on  to  a 


* Art.  124. 
t Arts.  123,  124. 

t Jour.  Inst.  Elect.  Eng.,  Vol.  21,  p.  677. 

§ Hay,  On  Impulsive  Current  Rushes  in  Inductive  Circuits,  London  Elec- 
trician, Vol.  33,  pp.  229,  277,  and  305. 


584 


ALTERNATING  CURRENTS 


circuit,  tlie  current  does  not  instantly  assume  the  final  form  of 
the  wave,  but  comes  gradually  to  its  final  form  through  a short 
interval  of  time.  The  length  of  the  interval  and  the  magnitude 
of  the  early  current  depend  upon  the  instantaneous  reactance 
of  the  circuit,  the  frequency,  and  the  point  in  the  voltage  wave 
at  which  the  connection  is  made.  The  instantaneous  reactance 
of  the  circuit  depends  upon  the  magnitude  of  the  residual  mag- 
netism in  the  core  at  the  instant  of  switching  in  the  current, 
and  its  direction  compared  with  the  instantaneous  impressed 
voltage  at  the  instant  of  switching  in.  If  the  instant  of  switch- 
ing on  to  the  circuit  is  that  at  which  the  impressed  voltage  is 
passing  through  zero,  the  current  in  the  transformer  is  less  dur- 
ing the  early  interval  than  its  final  value  ; while  if,  at  the  instant 
of  switching  on,  the  impressed  voltage  is  passing  through  its 
maximum  value,  there  may  be  quite  an  excess  of  current  flow 
through  the  circuit  for  a short  time,  on  account  of  the  relations 
which  exist  between  the  instantaneous  impressed  and  counter 
electric  voltages  during  the  first  half  period.  If  the  residual 
magnetism  has  a direction  in  the  core  corresponding  to  the 
magneto-motive  force  of  the  current  at  the  instant  of  switching 
in,  the  apparent  reactance  may  be  very  small,  and  the  instan- 
taneous current  rush  be  correspondingly  large ; but  if  the  resid- 
ual magne'tism  is  in  opposition  to  the  magneto-motive  force  of 
the  current  at  the  instant  of  switching  in,  the  quick  reversal  of 
this  magnetism  may  give  an  apparent  large  reactance  for  the 
instant  and  the  current  may  rise  gradually.  The  abnormal 
state  of  the  current  can  only  exist  for  a very  short  time  unless 
the  reactance  of  the  circuit  approaches  a condition  of  resonance. 

When  the  transformer  circuits  contain  resistance,  self-induct- 
ance, and  capacity,  the  disturbances  upon  opening  or  closing 
the  switching  devices  controlling  the  circuits  may  be  excessive 
and  may  cause  the  flow  of  very  large  momentary  currents  or 
the  generation  of  abnormal  voltages.*  Troublesome  effects  of 
this  nature  are  most  apt  to  occur  on  high  voltage  transmission 
lines  and  where  large  units  of  power  are  used.  The  excep- 
tional care  with  which  transformer  terminals  are  insulated  is  in 
part  necessary  on  account  of  the  unusual  conditions  that  exist 
upon  opening  or  closing  circuit  switches  or  upon  sudden  shifts 
of  load. 


* Art.  59. 


MUTUAL  INDUCTION,  TRANSFORMERS 


585 


147.  Methods  of  Testing  Transformers.  — The  commercial 
output  rating  of  a transformer  should  be  the  number  of  kilo- 
volt amperes  it  will  supply  to  a load  at  unity  power  factor, 
without  heating  beyond  a specified  rise  of  temperature,  when 
the  rated  full  load  voltages  are  used  in  the  primary  and  second- 
ary circuits,  and  when  the  voltage  and  current  waves  are  ap- 
proximately sinusoidal.  Commercial  efficiency  and  regulation 
have  already  been  defined.*  The  most  important  tests  of  trans- 
formers are  those  for  determining  the  temperature  rise  at  full 
load,  and  at  any  overload  for  which  the  transformer  is  de- 
signed ; the  regulation  between  no  load  and  full  load,  or  at  other 
proportions  of  load,  as  desired  ; the  core  losses  and  the  copper 
losses,  from  which  to  compute  the  efficiency  ; and  the  dielectric 
strength  of  the  insulation  between  the  primary  and  secondary 
coils  and  the  coils  and  core.  The  efficiency  and  regulation 
are  often  desired  also  for  loads  with  power  factors  other  than 
unity.  In  addition  to  tests  to  determine  those  quantities  it  is 
often  desirable  to  obtain  the  values  of  the  exciting  current 
under  various  conditions. 

It  is  evident  that  having  obtained  the  values  of  the  coil  re- 
sistances and  leakage  reactances,  the  iron  losses,  and  the  value 
and  power  factor  of  the  exciting  current,  the  more  important 
quantities  related  to  the  transformer  operation  except' the  heat- 
ing can  be  closely  determined  for  various  loads  and  power  factors 
by  the  use  of  the  transformer  circle  diagrams  and  formulas. 

All  tests  should  be  made  with  load  currents  and  voltages  of 
rated  value  and  frequency,  and  approximating  to  sinusoidal 
form,  that  is,  having  ordinates  not  varying,  at  any  instant, 
more  than  10  per  cent  from  that  of  the  equivalent  sinusoid  ; 
with  a room  temperature  at  25°  centigrade  or  with  the  quantities 
affected  properly  corrected  ; with  the  barometer  at  760  mm., 
or  with  proper  corrections  for  other  conditions ; with  the  rise 
of  temperature  of  the  transformer  at  the  normal  value  for  the 
conditions  of  the  test ; and  witli  the  apparatus  operating  under 
the  normal  conditions  for  which  it  has  been  designed  and  rated. 

In  this  country  the  Standardization  Rules  of  the  American 
Institute  of  Electrical  Engineers  are  generally  accepted  as  au- 
thoritative.! In  these  rules  are  given  detailed  statements  of  the 

* Arts.  123,  137. 

t Trans.  Amer.  Inst.  Elect.  Eng  , 1907,  pp.  1797-1825,  and  Year  Book,  ditto , 1909. 


586 


ALTERNATING  CURRENTS 


conditions  under  which  tests  of  the  electrical  apparatus,  in- 
cluding transformers,  should  be  made ; and  they  should  be  con- 
sulted before  making  tests  of  the  kind  indicated  above.  The 
following  few  paragraphs  give  in  outline  some  of  the  simpler 
methods  for  testing  transformers. 

Wattmeter  Method.  — A wattmeter  may  be  placed  in  the  pri- 
mary circuit  of  a loaded  transformer  and  another  in  the  second- 
ary circuit.  Then  the  ratio  of  their  readings  is  the  efficiency  of 
the  transformer.  Or,  a wattmeter  may  be  used  in  the  primary 
circuit,  and  an  amperemeter  and  voltmeter  can  be  used  to  deter- 
mine the  output  if  the  transformer  is  worked  on  a purely  non- 
reactive load,  though  it  is  always  wisest  to  use  a wattmeter 
on  alternating-current  circuits  for  measuring  power.  The  watt- 
meter was  early  used  by  Fleming  in  an  extended  series  of  trans- 
former tests,  and  found  to  be  satisfactory.  It  has  now  become 
the  standard  method  for  measuring  electrical  power.  Before 
taking  the  readings  the  transformers  should  be  run  at  full  load 
until  the  temperature  has  become  constant  in  all  parts,  which 
requires  from  live  to  fifteen  hours,  depending  upon  the  size.  Also, 
such  other  conditions  as  are  specified  earlier  in  the  article  and  in 
the  rules  referred  to  should  be  carefully  observed.  This  method 
requires  a supply  of  power,  and  a secondary  load  for  absorbing 
it,  equal  to  not  less  than  the  full  load  capacity  of  the  apparatus, 
or  larger  if  the  efficiencies  at  overloads  are  required.  When  very 
large  transformers  are  under  test  this  may  be  impossible  or  un- 
desirable, in  which  case  one  of  the  methods  given  later  may  be 
adopted.  In  transformer  manufacturing  establishments  it  is 
often  possible  to  feed  back  the  secondary  test  load  into  the  load 
circuits  of  the  works.  It  is  common  in  testing  to  obtain  the 
efficiency  at  |,  1,  1|,  and  sometimes  1-|  the  rated  output. 
For  great  accuracy  these  efficiencies  should  each  be  obtained  after 
the  temperature  has  become  constant.  It  is  also  not  unusual 
to  make  the  efficiency  tests  for  several  power  factors  if  the  trans- 
former is  to  be  used  on  loads  of  various  power  factors. 

Figure  813  shows  a tri-phase  transformer  connected  up  with 
tri-phase  wattmeters  in  the  primary  and  secondary  circuits 
as  required  for  making  an  efficiency  test,  though  ordinarily 
two  single-phase  wattmeters  would  be  used  in  place  of  each 
tri-phase  instrument.  On  the  low  voltage  load  side  the  watt- 
meter current  coils  are  attached  to  the  secondaries  of  current 


MUTUAL  INDUCTION,  TRANSFORMERS 


587 


Fig.  313.  — Tri-phase  Wattmeters  connected  to  a Tri- 
phase Transformer  for  Testing.  Connections  are  also 
shown  for  Current  and  Potential  Instrument  Trans- 
formers. 


transformers  (C.  T7.),  while  on  the  high  voltage  primary  cir- 
cuit the  instrument  voltage  coils  are  attached  to  the  secondaries 
of  potential  trans- 
formers (P.IZ7.)  in 
addition  to  the  cur- 
rent coils  being  con- 
nected to  current 
transformers.  In 
making  the  test, 
two  voltmeters  and 
amperemeters  would 
ordinarily  be  also 
connected  into  two  of  the  circuits  of  the  primary  and  secondary 
leads. 

The  method  here  described  has  the  disadvantage  of  depend- 
ing1 for  its  result  on  the  ratio  of  two  instrument  readings  that 
are  nearly  alike,  and  error  is  therefore  introduced  into  the 
efficiency  measurement  nearly  in  proportion  to  the  errors 
inherent  in  the  instrument  readings.  Additional  errors  are 
introduced  when  current  and  potential  transformers  are  used. 
This  method,  therefore,  should  only  be  used  for  purposes  of 
checking  results  of  other  methods  or  under  circumstances 
in  which  other  and  more  accurate  methods  are  not  avail- 
able. 

Stray  Power  Methods  for  obtaining  Efficiency.  — A very 
convenient  method  of  measuring  the  efficiency  of  transformers 
is  to  determine  the  various  losses  directly,  and  thence  the 
efficiency  by  calculation.  The  iron  losses  may  be  determined 
by  measuring  with  a wattmeter  the  power  absorbed  by  the 
transformer  when  the  secondary  circuit  is  open,  and  may  be 
considered  to  be  constant  for  all  loads  with  sufficient  accuracy 


for  commercial  purposes,  since  the  core 


magnetism 


changes 


but  little  with  changes  of  load  in  constant  voltage  trans- 
formers. The  copper  losses  for  any  load  are  readily  calculated 
when  the  secondary  and  exciting  currents  and  the  primary  and 
secondary  resistances  are  known.  The  exciting  current  may 
be  measured  at  the  same  time  that  the  iron  losses  are  de- 
termined, by  the  insertion  of  an  amperemeter  into  the  primary 
circuit  with  the  wattmeter ; and  the  resistances  may  be 
measured  by  a bridge  or  voltmeter  and  amperemeter  method 


588 


ALTERNATING  CURRENTS 


using  direct  currents.  For  a given  load,  the  secondary  current 
is  a fixed  quantity.  The  efficiency  is  then,  practically, 


P <1  + Pc  + 1*1  P%  + ( + In  ) Pi 


In 


where  Pc  represents  the  measured  iron  losses  and  s the  ratio  of 
transformation.  Due  care  must  be  used  that  the  temperatures 
are  right,  or  else  proper  corrections  must  be  made  to  the  coil 
resistances.  The  iron,  loss  will  vary  only  slightly  with  the 
temperature  and  need  not  ordinarily  be  corrected. 

A still  more  convenient  method,  which  may  be  readily  used 
in  central  stations  for  testing  transformers,  is  to  measure  the 
iron  losses  by  a wattmeter,  as  explained  above.  The  copper 
losses  may  then  be  measured  by  short-circuiting  the  secondary 
winding  through  an  amperemeter,  and  adjusting  the  primary 
voltage  until  the  full  load  current,  or  any  desired  fraction 
thereof,  passes  through  the  amperemeter.  The  reading  of  a 

wattmeter  in  the 
primary  circuit  is 
now  nearly  equal 
to  the  copper  losses 
for  the  tempera- 
ture of  the  wind- 
ings at  the  time, 
since  the  voltage 
and  maximum 
magnetic  density 
must  be  very  small,  and  the  iron  losses  are  therefore  almost  or 
entirely  negligible.  The  exact  copper  losses  may  be  determined 
by  measuring  and  correcting  for  the  small  iron  loss.  The  tests 
for  the  iron  losses  may  often  be  conveniently  made  by  using 
the  low  resistance  coil  of  the  transformer  as  the  primary  coil 


Fig.  314. — Transformer  arranged  for  measuring  Iron 
Losses.  Low  Voltage  Circuit  used  as  Primary. 


(Fig.  344). 

This  method  of  testing  may  be  used  with  satisfaction  where 
numerous  transformers  must  be  tested,  since  the  losses  and 
efficiency  may  be  determined  expeditiously  and  with  the  ex- 
penditure of  little  power.  When  combined  with  a run  of 
several  hours  with  full  load  current,  the  secondary  circuit 


MUTUAL  INDUCTION,  TRANSFORMERS 


589 


being  made  up  of  impedance  coils,  the  method  proves  econom- 
ical for  shop  tests. 

Ayrton  and  Sumpner  early  devised  a method  which  is  fre- 
quently used  in  which  two  transformers  of  the  same  size  and 
make  are  opposed  to  each  other,  and  which  is  often  called  the  op- 
position method.  The  method  of  connecting  is  shown  in  Fig.  345, 
in  which  A and  B are 
the  transformers  to 
be  tested,  with  their 
primary  windings 
connected  in  relatively 
opposite  directions  to 
the  leads  and  their 
secondary  windings 
in  series  with  each 
other.  The  voltages 
of  the  secondary 
windings  are  thus  in 
opposition.  A trans- 
former T is  inserted 
with  its  secondary 
winding  in  circuit 
with  the  secondaries 
of  A and  B.  By  vary- 
ing the  resistance  R, 
the  voltage  of  T may 
be  regulated  so  that  any  desired  currents  will  pass  through  the 
primary  and  secondary  windings  of  A and  B.  Then  the  output 
of  T measured  by  the  wattmeter  W2  will  give  approximate^  the 
copper  losses  of  A and  B plus  the  loss  in  leads  and  instruments. 
The  power  supplied  to  the  primary  circuits  of  A and  B by  the 
alternator,  measured  by  the  wattmeter  Wv  gives  approximately 
the  iron  losses  of  the  two  transformers.  From  the  data  thus 
derived  the  efficiencies  may  be  computed  on  the  assumption  that 
the  losses  are  equally  divided  between  the  transformers  A and 
B.  Various  modifications  of  this  method  may  be  made,  such 
as  putting  an  autotransformer  in  place  of  T,  or  the  two  primary 
circuits  may  be  fed  from  two  branches  connected  to  the  mains, 
but  with  the  voltage  of  one  regulated  above  normal  and  the  other 
below  normal,  so  that  the  difference  will  be  sufficient  to  send 


Fig.  345. — Transformers  connected  up  for  Testing  by 
the  Feeding  Back  or  Opposition  Method. 


590 


ALTERNATING  CURRENTS 


full  load  current  through  the  short-circuited  joint  secondary 
circuit.  Wattmeters  in  the  primary  branches  will  then,  together, 
measure  the  entire  losses  of  the  two  transformers.  An  ampere- 
meter should  he  inserted  in  the  primary  or  secondary  circuit 
by  which  to  regulate  the  current. 

Either  the  high  or  low  voltage  coil  may  be  used  as  the  pri- 
mary coil  in  making  transformer  tests,  as  is  convenient.  The 
last  method  described  is  largely  used  in  manufacturing  establish- 
ments or  when  two  transformers  of  similar  make  are  to  be  tested. 

It  must  be  borne  in  mind  that  the  maximum  magnetic  density 
in  the  core  which  corresponds  to  any  particular  impressed 
voltage  as  read  by  voltmeter  is  smaller  when  the  voltage 
wave  is  peaked  and  larger  when  the  voltage  wave  is  flat- 
topped,  than  when  the  wave  is  sinusoidal,*  and  the  core 
loss  is  therefore  less  for  a transformer  run  on  a circuit  with  a 
peaked  voltage  wave  than  when  the  voltage  wave  is  flat-topped, 
other  things  being  equal,  and  correction  must  be  made  for  that. 

Regulation  Tests.  — The  regulation  of  transformers  which  are 
used  in  incandescent  lighting  is  a matter  of  much  moment,  and 
regulation  tests  are  of  almost  equal  importance  to  the  tests  of 
losses  and  temperatures.  An  ordinary  method  of  making  regu- 
lation tests  is  to  place  a voltmeter  across  the  primary  circuit  and 
another  across  the  secondary  circuit  of  the  transformer  tobe  tested, 
care  previously  having  been  taken  that  the  transformer  is  at  the 
specified  temperature  and  other  conditions  called  for  in  the  test- 
ing rules  heretofore  referred  to  are  observed.  At  no  load,  the  re- 
duced equivalent  readings  of  the  instruments  should  be  equal,  and 
the  numerical  difference  between  the  reduced  readings  at  any 
other  load  gives  the  drop  of  voltage  corresponding  to  that  load. 
The  reduced  readings  are  gained  by  dividing  the  reading  of  the 
voltmeter  in  the  high  voltage  circuit  by  the  ratio  of  transforma- 
tion of  the  transformer.  This  method  lies  under  the  serious 
disadvantage  that  the  result  to  be  measured  is  a small  quantity 
equal  to  the  difference  of  the  much  larger  observed  quantities, 
and  the  ordinary  errors  inherent  in  instrumental  readings  there- 
fore are  likely  to  introduce  large  errors  in  the  value  found  for 
the  voltage  drop.  The  drop  of  voltage,  measured  in  this  way, 
includes  the  IR  drop  in  the  windings  and  the  drop  due  to 
magnetic  leakage,  both  of  which  increase  with  the  load.  The 

*Arts.  115,  134. 


MUTUAL  INDUCTION,  TRANSFORMERS 


591 


magnetic  leakage  drop  may  be  determined  by  subtracting  from 
the  total  drop,  the  value  of  the  IR  drop  which  is  calculated 
from  measured  resistances  and  currents.  The  scalar  value  of 
the  induced  leakage  voltage  is  obtained  from  the  expression 

e^-Vej-%? 

when  Ed  is  the  total  drop  measured  and  ER  is  the  total  drop  in 
the  secondary  voltage  due  to  the  resistance  of  the  two  coils. 
Having  obtained  EL  and  ER , and  the  exciting  current,  for  a 
given  voltage  and  load,  the  circular  transformer  diagram  loci 
may  be  drawn  as  earlier  explained.*  The  transformer  reactance 
equals 


A much  more  accurate  regulation  test  may  be  made  by  using 
two  transformers  of  equal  transformation  ratios  and  one  volt- 
meter. The  primary  circuits  are  separately  connected  to  the 
supply  mains,  and  the  secondary  circuits  are  connected  together 
on  one  side,  terminals  of  like  polarity  being  tied  together.  A 
voltmeter  of  high  resistance  or  an  electrostatic  voltmeter  is  con- 
nected between  the  other  legs  of  the  secondary  circuits.  The 
reading  of  this  voltmeter  with  a load  on  one  transformer,  when 
the  other  is  unloaded,  gives  the  total  drop  of  voltage  caused 
by  loading  the  former. 

Regulation  ratings  ai'e  usually  made  with  non-reactive  loads, 
though  the  regulation  for  loads  of  lower  power  factor  is  often 
very  important.  The  regulation  of  a transformer  is  changed  for 
the  worse  by  introducing  inductance  into  the  secondary  circuit, 
and  for  the  better  by  introducing  electrostatic  capacity  into  the 
secondary  circuit,  as  has  already  been  proved. f Regulation 
tests  on  a reactive  load  are  made  in  the  same  manner  as  ex- 
plained above.  The  power  factor  of  the  load  may  be  obtained 
by  dividing  the  wattmeter  reading  of  the  load  by  the  product 
of  the  secondary  amperes  and  volts. 

The  efficiency  and  regulation  of  polyphase  transformers  can 
be  obtained  by  the  same  methods  as  have  been  explained  for 
single-phase  transformer  tests,  using  the  methods  given  earlier 
for  measuring  polyphase  power  and  power  factor. | 

Heating. — The  rated  capacity  of  a transformer  is  determined 
* Arts.  130,  132.  t Art.  132.  t Art.  104. 


59  2 


ALTERNATING  CURRENTS 


mainly  by  the  amount  it  rises  in  temperature  clue  to  the  heat  caused 
by  internal  losses  of  power.  The  rise  of  temperature  is  affected 
by  the  temperature  of  the  surrounding  air,  the  barometric  condi- 
tions, and  extraneous  movements  of  air  over  the  apparatus.  There- 
fore these  conditions  during  a test  should  be  made  to  accord  with 
the  accepted  standards.*  The  temperature  of  the  conductors  can 
be  accurately  obtained  by  measuring  their  resistance  while  at  the 
temperature  of  the  room  in  which  the  test  is  being  conducted  and 
again  after  their  temperature  has  become  constant  under  load. 
The  change  of  temperature  can  then  be  determined  by  substitute 


ing  in  the  formula  ?2  - ^ when  a is  the  tempera- 

ture  coefficient  of  copper  with  respect  to  zero  degrees  centigrade 
and  may  be  ordinarily  taken  as  .0042,  R0  is  the  resistance  of  the 
windings  at  zero  degrees  centigrade  and  is  obtained  from  the 
formula  R0  = i21/(l  + at1  ),  Rx  is  the  resistance  measured  at 
the  initial  temperature  tv  t1  is  the  temperature  of  the  transformer 
when  the  first  resistance  measurement  was  made,  R2  is  the 
resistance  measured  at  the  final  tempei'ature,  and  f2  — t1  is  the 
rise  of  temperature  to  be  determined. 

The  temperature  of  core,  case,  and  insulation  can  be  measured 
by  thermometers,  as  can  also  that  of  the  windings  as  a check 
upon  the  resistance  temperature  measurement.  The  bulbs  of  the 
thermometers  must  be  carefully  protected  from  improper  radia- 
tion. In  transformers  with  water  circulation,  air-blast,  or  forced 
oil  circulation  the  temperatures  at  inlet  and  outlet,  and  the 
volumes  of  the  water,  air,  or  oil  respectively  introduced  per 
unit  of  time,  should  also  be  measured.  From  these  data  and  the 
specific  heats  of  the  cooling  agent  the  amount  of  heat  disposed 
of  by  the  special  cooling  device  may  readily  be  obtained.  In 
selecting  a transformer,  the  amount  of  heat  to  be  removed  by 
special  cooling  devices  should  be  specified.  No  part  of  the  trans- 
former should  exceed  50°  centigrade  rise  in  temperature  for  the 
full  load  it  is  to  carry,  and  from  5°  to  10°  lower  is  preferable, 
referred  to  an  initial  temperature  of  25°  centigrade. 

Dielectric  Strength  of  Transformer  Insulation. — The  dielectric 
strength  of  the  insulation  of  Transformers  is  tested  by  applying 
sufficiently  high  voltages  between  the  primary  coil  and  the  core 
or  the  secondary  coil  and  the  core  connected  together,  to  insure 


* Trans.  Amer.  Inst,  of  Elect.  Eng.,  1907,  pp.  1815-1818. 


MUTUAL  INDUCTION,  TRANSFORMERS 


593 


an  insulation  factor  of  safety  which  will  be  a sufficient  protection 
against  accident.*  For  transformers  in  which  the  high  voltage 
coil  is  for  550  to  5000  volts  the  accepted  testing  voltage  for  this 
purpose  is  10,000  volts  when  the  secondary  coil  connects  directly 
with  power  consumption  circuits.  The  general  rule  for  other 
transformers  is  that  the  test  voltage  shall  be  twice  that  of  the 
high  voltage  coil.  The  test  voltage  should  be  applied  for  the 
period  of  one  minute.  When  the  test  voltage  exceeds  10,000 
volts,  great  care  must  be  exercised  that  the  voltage  be  brought 
gradually  to  its  full  value,  and  while  being  maintained  no  large 
variations  occur  in  it,  as  otherwise  destructive  oscillations 
may  be  set  up. 

The  testing  voltage  may  be  measured  by  a voltmeter  in  con- 
nection with  a potential  transformer  if  necessary  or  by  means 
of  a standard  spark  gap.  Where  high  test  voltages  are  used, 
the  connections  must  be  made  with  great  care  to  prevent  a spark 
occurring  due  to  a break,  which  is  especially  apt,  on  account  of 
its  own  oscillatory  character,  to  cause  trouble.  For  this  same 
reason  the  spark  gap,  when  used,  should  be  in  series  with  a 
large  resistance.  The  testing  voltage  should  be  sinusoidal, 
under  which  conditions  the  stress  imposed  on  the  dielectric  is 
1.41  times  the  testing  voltage  observed  by  the  voltmeter.  The 
ordinary  commercial  frequencies  are  all  equally  effective  for 
testing  purposes. 

Special  testing  transformers  are  often  employed  for  testing 
the  dielectric  strength  of  machines.  These  are  designed  with 
relatively  small  electrostatic  capacity  and  large  impedance  which 
tends  to  reduce  danger  due  to  a breakdown  of  the  dielectric 
from  the  causes  named  above. 

Determination  of  Wave  Shape. — Various  methods  of  checking 
the  shape  of  voltage  and  current  waves  used  in  testing  have  been 
used.  One  of  the  oldest  of  these  is  by  means  of  a Contact 
maker  revolved  synchronously  with  the  generating  apparatus 
and  so  arranged  that  instantaneous  values  of  currents  and  volt- 
ages may  be  obtained  at  any  series  of  angles  of  advance  during 
a cycle  ; but  undoubtedly  the  most  convenient  and  satisfactory 
apparatus  to  use  for  the  purpose  is  the  Oscillograph.  Briefly, 
this  consists  of  a looped  ribbon  wire,  mounted  so  as  to  have  a 
relatively  very  high  natural  frequency  of  mechanical  vibrations, 
* Trans.  Amer.  Inst.  Elect.  Eng.,  1907,  pp.  1811-1815. 


594 


ALTERNATING  CURRENTS 


which  is  placed  within  a strong  constant  magnetic  field. 
Through  this  loop  is  passed  a current  proportional  at  each  in- 
stant to  the  current  or  voltage  wave  that  it  is  desired  to  trace. 
The  loop  is  then  influenced  by  a torque  proportional  at  each 
instant  to  the  value  of  the  current  within  it,  acting  on  one 
half  of  it  in  one  direction  and  on  the  other  half  in  the  opposite 
direction.  By  a minute  mirror  attached  at  one  edge  to  one 
strand  of  the  loop  and  the  other  edge  to  the  other  strand  of 
the  loop,  it  is  possible  to  cast  a beam  of  light  upon  a film 
or  screen  which  repeats  very  accurately  a tracing  of  the  wave 
required.  The  subject  of  curve  tracing  is  more  fully  discussed 
later.* 

Methods  used  in  Some  Historically  Important  Tests.  — The  con- 
tact maker  was  used,  in  the  early  days  before  reliable  watt- 
meters had  been  invented,  for  tracing  the  voltage  and  current 
curves  of  a transformer ; and  from  the  data  thus  obtained 
efficiency  and  other  characteristics  were  calculated.  Among 
the  historic  investigations  of  this  kind  were  those  by  Ryan 
and  Merritt,f  John  Hopkinson,|  Mordey,§  and  others.  A large 
calorimeter  was  sometimes  used  in  which  the  transformer  was 
immersed  for  the  purpose  of  obtaining  its  losses.  ||  The  power 
in  the  transformer  circuits  was  sometimes  measured  by  the  split 
dynamometer  or  combinations  of  the  three  voltmeters  and  three 
amperemeters  method. 

That  the  value  of  the  iron  losses  is  largely  independent  of 
the  load  carried  by  a transformer  was  first  conclusively  proved 
experimentally  by  Ewing.  The  same  thing  was  also  proved  by 
Fleming's  experiments.  Figure  346  is  plotted  from  one  of 
Fleming’s  tests  made  on  a transformer  of  4000  watts  capacity. 
The  ordinates  of  line  AB  represent  the  differences  of  the  power 
in  the  primary  and  secondary  circuits  as  measured  by  watt- 
meters. The  calculated  copper  losses  are  represented  by  the 
ordinates  of  the  line  CD.  The  difference  of  the  ordinates  of 
the  lines  AB  and  CD  at  any  point  is  the  iron  loss  for  the  par- 

* Art.  157. 

t Trans.  Amer.  Inst.  E.  E.,  Vol.  7,  p.  1. 

t Hopkinson’s  Dynamo  Machinery  and  Allied  Subjects,  p.  187  ; Elect.  World, 
Yol.  20,  p.  40. 

§ Jour.  Inst.  E.  E.,  Vol.  18,  p.  008. 

||  Duncan,  Electrical  World , Vol.  9,  p.  188 ; Roiti,  La  Lumiere  Electrique, 
Vol.  35,  p.  528. 


MUTUAL  INDUCTION,  TRANSFORMERS 


595 


ticular  load.  The  lines  AB  and  CD  are  approximately  paral- 
lel, which  shows  that  the  iron  losses  were  practically  constant, 
regardless  of  the  load.  Therefore  the  conclusion  was  drawn 
that  the  stray  power  methods  of  testing  transformers  give 
efficiencies  which  are  entirely  reliable. 


OUTPUT  IN  SECONDARY  WATTS 

Fig.  .'540.  — Curves  showing  Results  of  a Classic  Experiment  for  determining  Iron 
Losses  with  Varying  Load. 

Ewing’s  very  neat  plan  for  proving  this  point  was  designed 
to  get  at  the  matter  directly.  Two  small  transformers  were 
made  up  exactly  alike,  the  cores  of  which  consisted  of  insulated 
iron  wire  wound  into  the  form  of  a ring.  Over  this  were 
uniformly  wound  two  layers  of  wire  making  a primary  coil, 
and  another  two  layers  making  a secondary  coil.  In  operating, 
the  primary  and  secondary  coils  were  respectively  connected 
in  series,  but  the  two  halves  of  each  coil  in  one  transformer 
were  so  connected  as  to  be  in  magnetic  opposition  (Fig.  347). 
The  core  of  one  transformer  was  therefore  magnetized  and  that 
of  the  other  was  not.  While  the  I2R  losses  at  any  load  were 
equal  in  the  two,  the  transformer  with  magnetized  core  heated 
more  when  put  in  operation  than  the  other  transformer,  but 
the  temperature  of  the  second  was  brought  to  equality  with 
that  of  the  first  by  passing  a direct  current  through  the  in- 
sulated wire  of  the  core.  The  power  expended  in  heating  the 
second  core  by  the  direct  current  was  thus  equal  to  that  ex- 
pended in  the  first  core  due  to  iron  losses.  The  equality  of 
temperature  was  determined  by  means  of  thermo-electric 
couples  embedded  in  the  cores,  which  were  connected  in  series 
with  each  other  through  a galvanometer  (Fig.  347).  In  this 
experiment  it  was  found  that,  after  a balance  of  temperature 
was  once  obtained,  it  was  unaltered  by  any  changes  in  the 
loads  of  the  transformers,  thus  showing  that  the  core  losses 


596 


ALTERNATING  CURRENTS 


in  the  magnetized  transformer  were  independent  of  the 
load.* 

These  tests  fully  confirm  the  a priori  deduction  that  the  core 
loss  must  be  substantially  independent  of  load  in  an  ordinary 
constant  voltage  transformer,  which  follows  from  the  relations 
of  counter-voltage  to  impressed  voltage.  Neither  the  con- 
clusions nor  the  experiments  apply  to  transformers  in  tvhich 


Fig.  347. — Ewing’s  Apparatus  for  determining  Iron  Losses. 

IR  drop  or  leakage  reactance  drop  absorb  a considerable  part 
of  the  impressed  voltage,  however  ; because  in  them  the 
mutually  induced  voltage  and  therefore  the  maximum  magnetic 
density  in  the  core  must  vary  with  the  load  if  the  primary  im- 
pressed voltage  is  maintained  constant,  and  the  core  losses 
vary  when  the  maximum  magnetic  density  is  varied  provided 
the  cycles  per  second  remain  the  same. 

* Ewing  and  Klaassen,  Magnetic  Qualities  of  Iron,  London  Electrician , Vol. 
32,  p.  731. 


CHAPTER  XI 


SYNCHRONOUS  MACHINES  — ALTERNATORS.  MOTORS,  RO- 
TARY CONVERTERS,  FREQUENCY  CHANGERS 

148.  Losses  in  an  Alternator.  The  principal  internal  losses 
of  alternators  are  caused  by:  (1)  T2R  loss  in  the  conductors 
on  armature  and  field  magnet : (2)  Eddy  currents  in  armature 
cores  and  field  magnet : (3)  Eddy  currents  in  armature  con- 
ductors: (4)  Hysteresis  in  armature  cores:  (5)  Friction  of 
bearing  and  brushes,  and  air  friction,  often  called  Windage. 

In  well-designed  direct-current  dynamos,  the  pole  pieces 
usually  cover  not  less  than  two  thirds  of  the  armature  surface. 
In  alternators,  the  poles  usually  cover  about  one  half  of  the 
armature  surface,  or  a little  more.*  This  would  make  it  appear, 
upon  a superficial  examination,  that  the  field  ampere-turns, 
and  therefore  the  field  losses,  must  be  much  greater  in  the 
alternator.  However,  since  alternator  armatures  are  made 
proportionally  larger  in  diameter  (usually  exterior  to  a rotat- 
ing field  magnet)  in  order  to  give  space  for  the  windings 
and  to  avoid  excessive  magnetic  leakage,  the  proportional 
excitation  really  required  need  not  be  much  increased  when 
the  magnetic  circuit  is  well  designed.  In  the  same  way,  while 
not  much  more  than  one  half  of  the  armature  surface  is 
covered  with  wire  in  a single-phase  machine,  the  surface  for 
winding  is  made  much  larger  by  increasing  the  diameter, 
while  the  number  of  revolutions  per  minute  of  the  rotating 
part  is  not  much  reduced.  Consequently,  the  voltage  produced 
in  a given  length  of  conductor  is  commensurable  in  the  two 
classes  of  machines  running  at  equal  speed.  When  windings 
for  two  or  more  phases  are  wound  upon  the  armature,  the 
entire  surface  is  covered,  and  as  a result,  taking  into  account 
the  advantages  in  design  named  above,  polyphase  machines 
can  be  made  to  give  a larger  output  per  pound  of  material,  with- 
out excessive  heating,  than  is  usual  for  direct  current  machines. 

* Art.  21. 

597 


598 


ALTERNATING  CURRENTS 


The  fact  that  the  copper  is  usually  divided  among  more 
cores  increases  the  length  of  wire  on  alternator  field  magnets 
for  a given  magnetizing  power,  as  compared  with  the  field 
magnets  of  direct-current  machines.  On  the  other  hand, 
the  losses  may  be  brought  by  careful  designing  to  approach 
the  average  values  used  in  direct-current  machines.  For  large 
machines  running  from  200  to  several  thousand  kilowatts 
capacity,  the  combined  armature  and  field  copper  losses  in 
good  practice  gradually  reduce  to  a combined  value  of  between 
one  and  two  per  cent  for  the  very  large  sizes.  A certain  line 
of  5000  kilowatt  machines  built  by  a large  manufacturer  in  this 
country  has  a combined  I2R  loss  of  approximately  one  per  cent, 
while  a similar  line  of  500  kilowatt  machines  has  close  to  two 
per  cent  I2R  loss.  The  loss  is  about  equally  divided  between 
the  field  and  armature  windings. 

Eddy  currents  in  alternator  cores  are  apt  to  cause  greater 
loss  than  in  ordinary  direct-current  machines  unless  greater 
precautions  are  introduced  for  preventing  them.  Thus,  in 
the  armatures  of  direct  current  machines  of  considerable  capac- 
ity, the  magnetic  cycles  in  the  core  seldom  reach  twenty-five 
complete  periods  per  second.  On  the  other  hand,  the  com- 
mercial frequencies  for  alternating  currents  are  now  25  and 
60  periods  per  second.  The  number  of  magnetic  cycles  per 
second  in  alternator  armatures  is  evidently  equal  to  the  fre- 
quency of  the  induced  voltage  and  current.  Since  the  heating 
in  the  core  discs  which  is  caused  by  eddy  currents  is  pro- 
portional to  the  square  of  tire  induced  voltage  and  therefore  to 
the  square  of  the  number  of  magnetic  cycles  per  second,  it 
becomes  particularly  important  that  the  discs  composing  al- 
ternator armatures  be  thin  and  well  insulated  from  each  other. 
Therefore,  the  iron  oxide  composing  the  mill  scale  on  the  sur- 
face of  the  laminations  is  not  always  to  be  depended  upon  for 
insulation,  and  the  stampings  may  be  covered  with  an  insulat- 
ing enamel.  However,  when  some  of  the  newer  irons  giving 
low  losses  are  used,  the  oxide  proves  sufficient.  All  burrs 
caused  by  punching  the  discs  or  truing  the  surface  of  the  core 
should  be  carefully  avoided  or  removed. 

Eddy  currents  in  the  pole  pieces  are  felt  quite  severely  in 
some  types  of  alternators,  but  the  loss  caused  by  them  can 
always  be  brought  within  reasonable  limits,  in  well-designed 


SYNCHRONOUS  MACHINES 


599 


machines,  by  making  the  pole  faces  of  laminated  iron,  which 
is  cast  or  dovetailed  into  the  rotating  spider  frame.  Tufting 
of  the  magnetic  flux  in  the  polar  space  due  to  the  effect  of  the 
armature  teeth  is  apt  to  be  one  of  the  causes  of  eddy  currents 
in  the  pole  pieces.  This  trouble  can  be  reduced  by  careful 
design  of  the  teeth. 

The  loss  due  to  eddy  currents  in  armature  conductors  of  a fixed 
size  has  a tendency  to  be  greater  for  alternating  than  for  direct- 
current  dynamos,  on  account  of  the  more  frequent  and  sudden 
changes  in  the  strength  of  the  field  through  which  the  conductors 
pass.  However,  since  alternators  are,  in  general,  built  for  con- 
siderably higher  voltages  than  direct-current  machines  designed 
for  a similar  duty,  the  conductors  on  the  alternator  armatures 
are  of  proportionally  smaller  cross  section.  This  reduces  the 
relative  eddy  currents  to  such  an  extent  that  they  are  not  par- 
ticularly noticeable  except  in  very  large  or  special  machines. 
The  common  practice  of  winding  armature  coils  with  copper 
ribbons  or  bars  set  on  edge  (the  broad  side  parallel  with  the 
lines  of  force)  also  tends  to  decrease  eddy  currents.  In  very 
large  machines  built  to  generate  voltages  not  exceeding  2000 
volts,  armature  conductors  of  a large  cross  section  become  es- 
sential. In  such  cases  they  can  be  made  up  of  insulated  strips 
connected  in  parallel.  The  uniform  practice,  however,  of  build- 
ing alternator  armatures  with  embedded  conductors  avoids  all 
great  difficulty  from  eddy  currents  of  this  character,  since  the 
relatively  high  frequency  of  the  magnetic  cycles  in  the  core 
makes  it  undesirable  to  saturate  the  core  teeth  and  but  little 
stray  magnetism  from  the  field  magnets  gets  into  the  slots. 

The  effect  of  hysteresis  in  iron  core  armatures  is  proportional 
to  the  number  of  magnetic  cycles  per  second,  and  is  therefore 
usually  greater  in  alternator  armatures,  for  a given  magnetic 
density,  than  in  those  of  direct-current  machines.  There  are 
only  two  ways  of  decreasing  the  hysteresis  loss  per  cycle  and 
per  unit  volume:  (1)  by  reducing  the  magnetic  density;  (2)  by 
improving  the  quality  of  the  iron  used  in  the  core.  Reducing 
the  magnetic  density  serves  to  decrease  the  eddy  current  loss 
also,  and  is  therefore  doubly  advantageous.  With  the  best 
quality  of  ordinary  soft  steels,  the  magnetic  density  used  in 
the  cores  of  alternator  armatures  and  in  the  field  cores  varies 
commonly  from  10,000  to  12,000  lines  of  force  per  square  centi- 


600 


ALTERNATING  CURRENTS 


meter,  and  tends  toward  tlie  former  figure  for  machines  having 
frequencies  of  60  cycles  per  second.  In  the  teeth  the  density 
runs  up  to  from  15,000  to  sometimes  even  20,000  lines  per 
square  centimeter.  For  machines  of  25  cycles  frequency  the 
magnetic  densities  tend  to  mark  the  higher  values  given.  When 
the  new  silicon  and  other  alloy  steels,  which  have  very  low  hys- 
teresis constants,  are  used,  the  magnetic  densities  for  60-cycle 
machines  also  give  values  approaching  10,000  lines  of  force  per 
square  centimeter  or  slightly  more  in  all  parts  of  the  magnetic 
circuit.  In  the  case  of  these  steels  the  low  point  of  magnetic 
saturation  largely  determines  the  density  it  is  possible  to  use. 
Halving  the  magnetic  density  tends  to  quarter  the  eddy  current 
loss  for  an  equal  number  of  cycles  per  second.  In  the  same 
way,  the  hysteresis  loss  varies  nearly  in  the  proportion  of  B1S, 
or  in  other  words,  halving  the  magnetic  density  decreases  the 
hysteresis  loss  per  unit  volume  to  one  third.  On  the  other 
hand,  the  cross  section  of  iron  must  be  increased  to  decrease 
the  magnetic  density  per  square  centimeter.  This  makes  the 
careful  selection  and  handling  of  the  iron  designed  to  enter 
alternator  cores  of  special  importance.  The  total  iron  loss 
varies,  in  commercial  60-cycle  alternators  of  good  design,  from 
1.5  to  2 per  cent  of  the  rated  output  capacity,  and  this  is  in- 
creased about  .25  per  cent  for  alternators  having  a frequency 
of  25  cycles  per  second.  High  voltage  alternators  are  apt  to 
have  slightly  higher  iron  losses  than  those  for  low  voltage  on 
account  of  the  greater  depth  of  teeth  required  on  account  of 
the  additional  conductor  insulation. 

149.  Armature  Ventilation.  Density  of  Current  in  Conductors 
of  Alternators,  etc.  — Since  the  hysteresis  and  edd}'  current  losses 
are  apt  to  be  greater  in  alternator  armatures,  it  is  usually  nec- 
essary that  more  opportunity  be  given  for  cooling  than  is  the 
case  in  direct-current  machines.  The  real  effect  of  the  velocity 
of  rotation  upon  cooling  is  dependent  largely  upon  the  design 
of  the  rotating  field  magnet,  or,  if  the  field  magnet  is  stationary, 
upon  the  fanning  arrangements  made  in  the  construction  of  the 
armature;  but  for  a given  type  of  construction,  the  cooling 
effect  is  roughly  proportional  to  the  velocity,  as  was  early  indi- 
cated by  the  experiments  of  Rechniewski.*  Consequently  high 

* Bulletin  cle  la  Societe  Internationale  des  Electriciens,  1892 ; Electrical 
World , Vol.  19,  p.  336. 


SYNCHRONOUS  MACHINES 


G01 


surface  velocity  is  desirable  for  the  purpose  of  cooling.  Internal 
ventilation  of  rotating  armatures  is  also  rendered  more  effective 
by  I'eason  of  the  high  surface  velocity.  For  the  purpose  of  in- 
ternal ventilation  a revolving  armature  is  virtually  converted 
into  a centrifugal  blower,  which  sucks  in  air  at  its  center,  along 
the  shaft,  and  ejects  it  from  the  surface.  For  this  purpose  the 
openings  in  the  spider  which  supports  the  core  communicate  with 
the  surface  through  radial  ducts,  as  seen  in  Figs.  78  and  92. 
The  rotation  of  the  armature  then  causes  a continuous  circula- 
tion of  air  through  the  ducts,  which  is  proportional  in  volume  to 
the  surface  velocity  of  the  armature.  Sometimes  special  wings 
or  projections  are  placed  on  a rotating  armature  which  greatly 
assist  in  ventilating  the  field  magnet  and  also  the  armature. 
These  remarks  apply  with  particular  force  to  rotary  converters, 
which  always  have  rotating  armatures.  When  the  field  magnet 
revolves,  which  is  the  case  in  all  large  size  modern  alternators,  the 
poles  cause  a vigorous  circulation  of  air,  which  serves  in  lieu  of 
the  blower  action  of  a revolving  armature,  except  in  the  case  of 
steam  turbine  driven  field  magnets,  which  are  generally  built 
without  projecting  poles  so  as  to  give  a cylindrical  surface,  and 
such  field  magnets  are  provided  with  blower  ducts  similar  in 
character  to  those  of  revolving  armatures.  The  stationary 
armature  has  radial  ventilation  ducts  between  the  laminations, 
from  ^ to  inch  in  width  and  spaced  from  1|-  to  2|-  inches  apart. 
When  the  field  poles  are  properly  arranged,  large  volumes  of 
air  are  forced  through  the  ducts.  Figures  80,  81,  and  82  show 
the  ventilating  ducts  of  stationary  armatures  quite  clearly, 
while  Fig.  113  shows  the  adaptability  of  the  rotating  crown  of 
poles  on  a rotating  field  magnet  to  act  as  a fan.  The  blower 
ducts  of  a steam  turbine  driven  field  magnet  ordinarily  deliver 
air  into  corresponding  ducts  of  the  stationary  armature  of  the 
machine.  The  peripheral  velocity  of  rotating  field  magnets 
varies  from  3000  to  20,000  feet  per  minute,  the  latter  figure 
being  marked  in  certain  high  speed  alternators  of  large  capac- 
ity, while  the  velocities  near  the  lower  limit  are  suited  to  low 
speed  small  machines. 

With  all  precautions  to  avoid  excessive  heating  in  alternator 
cores,  careful  design  is  demanded  to  prevent  them  from  heating  to 
a higher  temperature  than  is  desirable,  especially  when  they  are  of 
the  modern  high  speed  types.  It  then  becomes  a matter  of  some 


002 


ALTERNATING  CURRENTS 


concern  to  determine  the  possible  amount  of  energy  which  may 
be  expended  in  the  armature  conductors  without  placing  an  ex- 
cessive additional  burden  upon  the  cooling  surface.  There  is 
no  valid  reason  for  admitting  a higher  temperature  limit  in  al- 
ternator armatures  than  is  allowed  in  direct-current  machines, 
as  was  formerly  done,  and  consequently,  the  radiating  surface 
per  watt  of  the  power  to  be  dissipated  should  be  the  same,  where 
the  ventilation  and  construction  are  similar.  On  account  of 
the  large  amount  of  heat  caused  by  core  losses  which  must  be 
radiated,  it  is  not  safe  to  allow  an  armature  I2R  loss  of  one 
watt  for  each  square  inch  of  cooling  surface.  In  common  prac- 
tice, for  each  watt  of  PR  loss  an  average  of  from  1|-  to  2 
square  inches  of  outside  winding  surface  is  allowed.  This 
constant  is  sometimes  made  much  larger,  but  seldom  smaller. 
This  makes  the  total  watts  to  be  radiated,  about  2 to  6 watts 
per  square  inch  of  the  surface. 

It  will  be  appreciated  that  the  velocity  of  an  alternate!’  not 
only  determines  the  amount  of  blower  action  possible  in  a 
machine  of  given  type,  but  also  determines  the  form  of  the 
masses  of  material  from  which  the  heat  must  be  extracted. 
Thus,  in  a low  speed  alternator  arranged  for  direct  connection  to 
a reciprocating  engine,  the  diameter  of  the  armature  must  be 
made  large  relatively  to  its  length  in  the  direction  of  the  shaft. 
While  in  a high  speed  machine,  arranged  for  connection  to  a 
steam  turbine,  the  diameter  of  the  armature  is  relatively  less  and 
its  length  greater.  This  difference  is  caused  by  the  different 
numbers  of  poles  required  and  the  permissible  circumferential 
velocity.  Further,  the  high  speed  machines,  up  to  certain 
limits  of  speed,  as  shown  by  Fig.  348*,  have  a better  weight 
efficiency  and  thus  have  a less  volume  of  material  per  watt 
loss. 

The  density  of  current  in  conductors  wound  upon  iron  core 
armatures  in  good  practice  should  usually  not  much  exceed  one 
ampere  for  from  300  to  1000  or  more  circular  mils  cross  sec- 
tion. The  density  of  current  that  is  allowable  in  either  arma- 
ture or  field  windings  varies  quite  widely  with  the  dimensions 
of  the  coils  and  the  efficiency  of  their  ventilation. 

Since  the  field  cores  of  alternators  are  usually  quite  thin,  the 
windings  are  often  of  a depth  equal  to  as  much  as  one  half 
*Behrend,  Electrical  Review , Yol.  45,  p.  375  et  seq. 


SYNCHRONOUS  MACHINES 


603 


the  thickness  of  the  cores.  At  the  same  time  this  depth  is 
generally  no  greater  than  that  on  many  direct  current  dynamo 
field  magnets.  The  same  radiating  constant  can  therefore  be 
safely  used,  that  is,  from  .35  to  .40  watt  per  square  inch  of  the 
outer  surface  of  the  winding  when  the  armature  rotates  as  in  ro- 
tary converters,  and  1.5  to  3 watts  per  square  inch  when  the  field 
magnet  rotates  — the  latter  value  depending  upon  the  speed  and 


SPEED  IN  REVOLUTIONS  PER  MINUTE 

Fig.  318.  — Relation  of  Speed  to  Weight  in  Alternators,  for  a line  of  1000  Kilowatt, 
3-Phase,  25-Cycle  Alternators. 


design  of  the  field  magnet.  The  ratio  of  winding  depth  to  thick- 
ness of  core  is  widely  variable,  and  the  radiation  constant  can 
wisely  be  based  upon  the  external  surface  of  the  windings.  The 
field  cores  should  therefore  be  made  of  sucb  a length  that  the  area 
of  the  external  surface  of  the  windings  on  rotating  field  magnets, 
in  square  inches,  may  be  from  ^ to  f times  the  PR  loss  in  watts. 
The  form  of  the  field  magnet,  and  therefore  the  effect  of  the 


604 


ALTERNATING  CURRENTS 


iron  in  conducting  away  the  heat  in  the  field  windings,  has  a 
considerable  effect  on  the  allowable  value  of  the  constant,  as  is 
also  the  case  in  direct-current  machines.  If  the  winding  depth 
exceeds  two  inches,  it  is  likely  to  cause  injurious  heating  in  the 
inner  layers.  The  field  windings  are  often  made  of  edgewise- 
wound  coils  of  wide,  thin  copper  strips.  Strips  of  insulation 
are  wound  with  the  copper  strips,  and  the  edges  of  the  copper 
strips  are  allowed  to  remain  bare.  This  type  of  construction  is 
• not  only  mechanically  strong  but  affords  ready  radiation  of  heat 
from  the  coils. 

The  cross-sectional  area  of  the  field  cores  is  given  by  the 

formula,  where  <I>  is  the  useful  armature  flux  re- 

quired  from  one  pole,  v is  the  leakage  coefficient,  and  B the 
magnetic  density  desired  in  the  field  core.  The  pole  width  is 
determined  by  the  mechanical  and  electrical  conditions  which 
fix  the  pitch,  and  only  the  length  of  the  pole  faces  may  be  al- 
tered to  vary  B.  The  value  of  the  coefficient  of  magnetic  leak- 
age is  quite  large  in  most  types  of  alternators.  It  probably 
averages  from  1.15  to  1.45  in  standard  alternators  with  poles  set 
in  a circle  either  without  or  within  the  armature.  The  calcula- 
tion of  this  coefficient  of  alternators  may  be  carried  out  upon  the 
same  methods  as  those  used  for  continuous  current-machines,* 
or  it  may  be  measured  for  a complete  machine  by  the  search  coil 
method;  that  is,  a test  coil  may  be  arranged  about  a pole  piece 
and  connected  to  a ballistic  galvanometer.  When  the  field 
current  is  then  suddenly  reversed,  the  throw  of  the  needle  is 
closely  proportional  to  the  magnetic  flux  in  the  pole.  A coil 
of  equal  turns  may  then  be  laid  on  the  armature  surface  inclos- 
ing an  arc  equal  to  the  polar  pitch  and  extending  the  full 
length  of  the ‘armature  coils.  The  throw  is  now  proportional 
to  the  useful  flux  from  the  pole.  The  ratio  of  the  first  to  the 
second  reading  is  approximately  proportional  to  the  leakage 
coefficient.  The  coefficient  varies  somewhat  with  different 
armature  currents  and  power  factors  on  account  of  the  mag- 
netic reactions  of  the  armature  currents  upon  the  poles. 
By  likewise  using  test  coils  placed  around  different  parts  of 
the  field  core,  the  distribution  of  the  magnetic  flux  may  be 
measured. 

* Jackson’s  Electro-magnetism  and  the  Construction  of  Dynamos,  p.  133. 


SYNCHRONOUS  MACHINES 


605 


150.  Determination  of  the  Number  of  Armature  Conductors. 

— The  formula  for  voltage  induced  in  an  alternator, 


F 2 KSQVp 
108  x 60  ’ 


may  be  put  into  the  form 

yr  _ 2 KS<l>f 
108  ’ 


since 


Vp 

60 


is  equal  to  the  frequency  f of  the  induced  voltage.* 


Taking  a proper  value  for  K,  as  already  explained,  gives  the  effec- 
tive value  of  E.  In  a well-designed  machine  of  the  usual  Ameri- 
can types,  the  value  of  K is  about  1.1,  which  is  the  value  for  a 
machine  which  gives  a sinusoidal  voltage  and  in  which  the  dif- 
ferential action  is  negligible.  The  conditions  of  service  usually 
fix  the  values  of  E and/ in  any  particular  case,  and  the  equation 
then  contains  two  dependent  variables,  <E>  and  S.  The  ratio 
of  these  is  determined  from  the  form  and  dimensions  of  the  ar- 
mature which  it  is  desired  to  make.  The  number  of  poles  is 
limited  by  constructive  considerations  and  the  importance 
of  keeping  the  magnetic  leakage  within  reasonable  limits.  On 
the  latter  account  the  poles  must  not  come  too  near  together. 
With  this  in  view  the  frequency  cannot  be  increased  beyond  a 
certain  limit  by  increasing  the  number  of  poles,  without  carry- 
ing the  periphery  velocity  of  the  armature  beyond  the  safe  limit. 
Thus,  suppose  a machine  is  designed  with  a twenty-pole  rotat- 
ing field  magnet,  and  its  field  magnet  is  designed  to  be  driven 
at  the  safe  limit  of  velocity.  If  it  should  be  desired  to  increase 
the  frequency  of  the  voltage  by  10  per  cent,  two  poles  must  be 
added  to  the  field  magnet  and  the  revolutions  kept  constant,  or 
else  the  field  magnet  must  be  speeded  up  10  per  cent.  Suppose 
the  latter  is  not  permissible  on  account  of  mechanical  safety; 
then,  if  the  poles  are  already  as  close  together  as  economy 
admits,  in  order  to  increase  the  number  of  poles  the  pitch  circle 
must  be  increased  in  diameter.  This  again  calls  for  an  in- 
crease of  the  periphery  velocity  of  the  field  magnet,  the  revolu- 
tions per  minute  remaining  constant.  Hence  in  any  type  of 
machine  a limit  of  frequency  may  be  reached  which  cannot  be 
safely  exceeded.  The  limiting  frequencies  in  the  ordinary  types 


* Art.  20. 


GOG 


ALTERNATING  CURRENTS 


of  machines  designed  for  commercial  service  are  from  100  to 
150  periods  per  second.  In  the  old  Mordey  type  the  limit  was 
much  higher,  since  the  poles  of  this  machine  may  be  very  close 
together  and  cause  no  additional  leakage,  and  structural  con- 
derations,  only,  set  the  limit.  Commercial  frequencies  for  elec- 
tric lighting  and  power  are  all  less  than  150  periods  per  second, 
and  are  usually  60  or  less,  so  that  all  practical  types  of  alter- 
nators may  be  used  in  commercial  service.  The  requirements 
of  wireless  telegraphs,  on  the  other  hand,  are  now  making  a 
demand  for  some  relatively  small  alternators  of  special  t}rpes 
designed  for  generating  currents  of  frequencies  as  great  as  two 
or  three  hundred  thousand  cycles  per  second. 

Fixing  the  frequency  of  a machine  and  the  periphery  velocity 
and  revolutions  of  the  revolving  part,  fixes  both  the  diameter 
of  the  armature  and  the  number  of  field  poles.  The  diameter 
of  the  armature  fixes  the  space  for  armature  slots.  With  the 
slot  space  fixed  and  the  dimensions  of  the  slots  determined 
from  the  conditions  of  tooth  saturation  and  core  reluctance 
desired,  the  value  of  iS  is  determined  from  the  number  of 
insulated  conductors  of  requisite  area  which  can  be  properly 
placed  in  the  slots.  The  value  of  S being  determined,  the 
length  of  the  armature  must  be  made  such  that  the  necessary 
total  magnetization  may  be  set  up  in  the  field  cores  and 
armature  without  forcing  the  magnetic  density  in  the  teeth 
and  cores  to  too  high  a value.  Finally,  the  ratio  of  radiating 
surface  to  I2M  loss  should  be  checked. 

It  is  well  to  consider  here  the  effect  upon  the  output  of  an 
alternator,  of  making  a change  in  the  number  of  armature  con- 
ductors. The  voltage  developed  in  the  coils  due  to  their  cut- 
ting lines  of  force  is  proportional  to  the  number  of  turns  in  the 
coils.  The  voltage  of  self-induction,  on  the  other  hand,  is  pro- 
portional to  the  square  of  the  number  of  turns  in  a coil.  Hence, 
increasing  the  number  of  armature  turns  beyond  a certain  limit 
may  actually  decrease  the  output  of  the  machine,  and  if  carried 
to  a sufficient  degree  may  make  it  even  tend  to  regulate  for 
constant  current  instead  of  constant  voltage.  The  following 
argument,  in  which  is  approximately  determined  the  number  of 
turns  that  will  give  maximum  output,  is  of  some  value  in  giv- 
ing a conception  of  the  effect  of  the  number  of  turns  upon 
output.  In  the  discussion  the  effect  of  armature  reactions  on 


SYNCHRONOUS  MACHINES 


607 


the  field  is  assumed  to  be  included  in  the  inductive  reactance 
of  the  winding.  Where  the  load  is  non-reactive  the  assumption 
is  justifiable.*  If  Ll  represents  the  effective  or  working  self- 
inductance for  each  armature  conductor,  including  its  propor- 
tion of  reactive  effect  on  the  field  magnet,  then  the  total 
effective  self-inductance  of  the  armature  is  approximately 
L—  S2LV  In  the  same  way,  if  E1  represents  the  voltage  de- 
veloped per  conductor,  the  total  voltage  developed  in  the  arma- 
ture is  E — SEV  From  the  fundamental  formula 


VR2  + 4 7 r2f2E2' 
we  get  EE1  = E2  - 4 7 j2f2EL\ 

or  IR  = VE2-  4 EPEE1. 


This  may  be  put  into  the  form 

IR  = V S2]lf-  4 EP1  WE2, 

supposing  the  external  circuit  to  which  the  alternator  is  con- 
nected to  be  non-reactive.  The  term  IR  is  the  active  voltage 
in  the  complete  alternator  circuit,  and  it  is  desirable  that  this 
be  a maximum  for  a given  armature.  Differentiating  the  equa- 
tion with  respect  to  S and  solving  for  a maximum,  we  get  the 
following  : 

d(IR ) = 2 S(E2-  8 EfEEL2)  = Q . 

dS  2 VS2E  2 - EEfU^L2 

hence  E j2  — 8 Ef2I2L12S2  =0.  Or  IR  is  a maximum  when 


S = . 

2 V2  7 rfILl 

In  this  it  is  assumed  that  the  armature  resistance  is  small  com- 
pared with  that  of  the  external  circuit,  which  is  always  the 
case  in  efficient  working. 

For  E1  may  be  substituted  its  value 


E,= 


2 K<i>  f 
~ 108_  ’ 


and  the  expression  for  S at  maximum  output  becomes 

a-  K*  - I-a, 

10«  tt/£,  ILi 

* Art.  155. 


608 


ALTERNATING  CURRENTS 


where  A is  a constant  depending  upon  the  type  and  dimensions 
of  the  machine.  If  K has  a value  of  1.1,  the  value  of  A is 
practically  25  x 10  ~10. 

The  final  form  of  the  variable  portion  of  the  expression  giving 
the  maximum  economical  value  of  S is  striking.  Its  numerator 
is  the  total  useful  magnetization  due  to  the  field  magnet  which 
passes  through  an  armature  coil,  and  its  denominator  is  the 
magnetization  passing  through  the  coil  due  to  the  current  in 
each  of  its  own  conductors.  The  fact  is,  however,  that  mechan- 
ical limitations  and  limitations  due  to  heating  and  regulation, 
and  the  considerations  of  efficiency,  prevent  the  number  of  con- 
ductors in  modern  alternator  armatures  even  approaching  the 
number  given  by  this  formula,  when  the  machine  is  worked  on 
a non-reactive  load. 

When  the  external  circuit  is  inductive,  as  is  commonly  the 
case,  the  number  of  armature  turns  which  gives  a maximum 
active  voltage  is  smaller  than  when  the  external  circuit  is 
non-reactive.  If  S'  is  the  number  of  conductors  giving  the 
maximum  active  voltage  when  the  external  circuit  has  self- 
inductance _LC,  and  S is  the  number  of  conductors  giving  the 
maximum  voltage  when  the  external  circuit  is  non-reactive,  then 


In  the  latter  expression  S2L}  is  the  self-inductance  of  the  arma- 
ture when  wound  with  the  proper  number  of  turns  to  give  a 
maximum  active  voltage  when  the  external  circuit  is  non- 
reactive.  This  becomes  larger  when  there  is  an  inductive 
load,  since  the  reactions  of  the  armature  upon  the  field  are 
greater.*  When  Le  is  greater  than  S2LV  the  right-hand  mem- 

S' 

her  of  the  expression  for  — becomes  imaginary ; that  is,  the 
ratio  is  an  indeterminate. 

It  is  impossible  to  put  so  many  turns  on  commercial  alter- 
nator armatures  as  the  criterion  shows  would  give  the  greatest 
output  at  unity  power  factor,  since  the  question  of  regulation 


or 


* Alt.  152. 


SYNCHRONOUS  MACHINES 


609 


in  constant- voltage  alternators  demands  that  the  fall  of  voltage 
in  the  armature  due  to  resistance  and  inductance  shall  be  as 
small  as  possible.  Nevertheless  where  alternators  are  subject 
to  inductive  loads,  the  self-inductance,  reactive  effect  upon  the 
field,  and  resistance  of  the  armature  conductors  are  most  im- 
portant elements  affecting  regulation  under  the  various  operat- 
ing conditions  to  be  met  in  practice,  as  is  shown  clearlj"  by  the 
relations  just  developed. 

151.  Armature  Ampere-turns  and  Self-inductance  of  Alterna- 
tors. — The  self-inductance  of  an  alternator  armature  may  be 
approximately  estimated  from  the  magnetic  and  electric  data 
of  the  machine.  The  reluctance  of  each  magnetic  circuit  must 
be  calculated  exactly  as  in  the  case  of  a direct-current  multipolar 
dynamo,  — in  order  to  determine  the  field  windings,  — using 

the  formula  P = where  l is  the  length,  A the  cross  section, 
/j.A 

and  /i  is  the  permeability  of  the  part  of  the  magnetic  circuit 


Fig.  349. —Diagram  for  showing  the  Magneto-motive  force  required  for  Each  Field 
Core  to  set  up  the  Required  Magnetic  Flux. 


under  consideration.  The  reluctance  to  be  overcome  by  the 
magneto-motive  force  of  each  field  core  belongs  to  that  part  of 
the  magnetic  circuit  which  lies  between  the  planes  parallel  to  the 
armature  shaft  in  which  lie  the  lines  AA'  and  BB'  in  Fig.  349, 
which  is  a section  of  an  alternator  with  rotating  armature. 
Machines  with  rotating  field  magnets  may  be  treated  in  the 


610 


ALTERNATING  CURRENTS 


same  manner.  Calling  that  reluctance  P,  the  ampere-turns 
for  each  field  core  are  in  number, 


nI=^-P 


, <£  P 

L _|_  8 3 _|_ 


1.25  1.25 


$ P cb  P 

1.25  1.25 


where  ®f,  <t>8,  <!>„,  and  <!>,,  and  Pf,  P8,  Pa,  and  Pt  are  respectively 
the  magnetic  fluxes  and  reluctances  of  the  field  core  and  yoke, 
air  gap,  armature  body,  and  armature  teeth  lying  between  AA' 
and  BB'. 

The  reluctance  in  the  different  parts  of  the  magnetic  circuit 
met  by  lines  of  force  which  are  set  up  by  the  magneto-motive 
force  of  the  armature  ampere-turns  when  the  field  magnet  is  ex- 
cited, may  be  assumed  to  be  equal  to  the  reluctance  in  the  same 
parts  of  the  circuit  met  by  the  lines  set  up  by  the  field  coils,  except 
for  the  part  of  the  flux  due  to  armature  turns  which  leaks  from 
tooth  to  tooth  in  a multi-tooth  armature.  The  number  of  Hues 
of  force  set  up  in  the  portion  of  the  magnetic  circuit  between 
the  planes  A A'  and  BB'  by  a unit  current  in  one  armature  coil, 


the  center  of  which  is  directly  under  a pole  face,  is 


1.25  S, 
2 P ’ 


where  Sx  is  the  number  of  conductors  in  the  coil  and  is  equal 
to  twice  the  number  of  turns  in  the  coil,  and  P is  the  equiva- 
lent combined  reluctance  of  the  magnetic  path  in  the  field  mag- 
net, armature,  and  air  gap.  Each  one  of  these  lines  of  force 
in  completing  its  circuit  through  another  segment  of  the  mag- 
netic circuit  must  link  another  armature  coil,  so  that  we  may 
say  that  the  number  of  lines  of  force  set  up  by  each  pair  of 


coils  is 


2 P 


The  self-inductance  of  the  pair  of  coils,  there- 


fore, so  far  as  relates  to  the  magnetic  circuit  through  the  field 
core,  is 


1.25  S? 

2 x P x 108’ 


since  Sx  is  equal  to  the  number  of  turns  in  two  coils. 

The  self -inductance  per  phase,  of  the  whole  armature  due  to 
this  magnetic  flux,  is  equal  to  Lx  multiplied  by  the  number  of 
pairs  of  coils,  when  the  armature  windings  of  a phase  are  con- 
nected in  series,  or 

L = 


1.25  Sfp 


SYNCHRONOUS  MACHINES 


611 


when  the  coils  are  placed  in  one  slot  per  pole  per  phase.  When 
the  windings  are  distributed  in  several  slots  per  pole  per  phase 
the  value  of  L is  diminished.  When  the  armature  is  connected 
with  the  halves  in  parallel,  this  inductance  is  one  fourth  as  great 
as  is  given  by  this  formula;  but  in  the  case  of  two  similar  arma- 
tures built  for  the  same  voltage  and  output,  the  one  connected 
with  the  halves  in  parallel  has  twice  as  many  conductors  in 
each  coil  as  has  the  other  armature,  and  their  self-inductances 
are  equal. 

The  path  of  the  lines  of  force  set  up  by  the  armature  coils 
has  been  assumed  to  be  the  same  as  the  path  of  those  set  up  by 
the  field  magnets,  and  the  real  effect  of  the  armature  magneto- 
motive force  upon  the  number  of  lines  of  force  in  the  mutual 
magnetic  circuits  is  to  increase  or  decrease  the  number  that 
would  exist  were  the  armature  turns  absent,  rather  than  to  set 
up  an  independent  magnetic  flux.  The  extent  of  this  effect 
depends  upon  the  lag  of  current  in  the  armature,  and  the  effects 
of  armature  reactions  and  of  self-inductance  are  therefore 
closely  related.  In  the  case  of  machines  with  toothed  armature 
cores,  as  already  said,  the  reluctance  measured  around  the  path 
of  the  leakage  lines  of  force  set  up  by  eacli  group  of  conduc- 
tors in  a slot  may  be  quite  small.  This  is  due  to  the  effect  of 
the  leakage  from  tooth  to  tooth.  Consequently,  the  self-induct- 
ance of  the  armature  must  be  increased  over  the  amount  given 
in  the  preceding  formula  by  an  amount  equal  to  that  caused  by 
the  tooth  to  tooth  leakage.  This  is  made  up  of  the  self-induct- 
ance caused  by  this  leakage  for  each  group  of  conductors  in  a 
slot  multiplied  by  the  number  of  groups  in  the  whole  winding 
of  a phase.  It  is  comparatively  constant  in  value  for  different 
armature  currents,  since  the  leakage  path  may  be  partially  in 
air,  and  the  teeth,  because  of  their  small  cross  section  relative  to 
other  parts  of  the  magnetic  field  circuit,  are  apt  to  be  magnetized 
to  a comparatively  high  degree  of  saturation. 

152.  Armature  Reactions  in  Alternators.  — The  armature  re- 
actions of  alternators  do  not  cause  as  serious  consequences  in 
some  respects  as  those  of  direct  current  machines,  as  the  com- 
mutator is  absent,  but  their  effect  upon  regulation  is  of  prime 
importance  and  demands  first  attention  from  the  designer. 
Consider  first  the  conditions  which  exist  in  single-phase  ma- 
chines, or  in  polyphasers  when  the  armature  currents  are 


612 


ALTERNATING  CURRENTS 


unbalanced,  which,  as  will  be  seen  later  in  the  discussion,  pro- 
duces much  the  same  effect  as  is  produced  in  single-phasers. 
Then  when  the  current  of  the  armature  is  in  exact  phase  with 
the  impressed  voltage,  the  armature  current  has  comparatively 
little  opportunity  to  affect  the  field  magnetism.  When  the  arma- 
ture conductors  are  directly  between  the  pole  pieces,  the  instan- 
taneous current  is  zero,  and  therefore  at  this  point  the  armature 
has  no  effect  upon  the  field  magnetism.  When  the  coils  have 


the  armature  toward  the  right-hand  with  respect  to  the  field  poles. 

moved  through  one  half  the  pitch,  a sheet  of  current  at  its  max- 
imum value  flows  directly  under  the  pole  faces.  This  current 
has  such  a direction  that  its  magnetic  effect  tends  to  crowd  the 
lines  of  force  of  the  field  magnet  into  the  trailing  tips  of  the 
poles  (Fig.  350).  Hence  the  field  is  weakened  on  account  of 
the  increased  reluctance  of  the  magnetic  circuit.  This  effect  is 
probably  not  very  marked  in  the  usual  forms  of  alternators,  since 
the  armature  ampere-turns  are  usually  not  large  compai’ed  with 
the  impressed  magneto-motive  force  of  the  field  magnets,  but,  to 


SYNCHRONOUS  MACHINES 


613 


whatever  extent  the  effect  exists,  it  produces  a periodic  shifting 
of  the  magnetic  density  across  the  pole  face  as  the  armature  coils 
move  from  pole  to  pole.  The  distortion  and  consequent  weak- 
ening of  the  field  may  be  reduced  by  cutting  a slot  longitudi- 
nally across  the  pole  faces,  or  by  carefully  designing  the  pole 
tips.  In  Fig.  350  the  irregular  line  dd  indicates  the  tendency 
of  the  groups  of  conductors  in  a slot  to  distort  the  curve  bb  on 
account  of  the  fact  that  the  magneto-motive  forces  of  their 
currents  are  not  evenly  distributed  over  the  armature  surface. 
This  effect  is  usually  not  large. 

When  the  armature  current  is  out  of  phase  with  the  induced 
voltage,  the  conditions  are  quite  different.  Suppose  that  the 
phase  of  the  current  is  retarded  on  account  of  self-induction. 
When  the  centers  of  the  coils  are  under  the  poles,  the  current 
is  not  zero,  but  has  an 
instantaneous  value 
which  depends  upon 
the  amount  of  retard- 
ation. This  current 
in  a generator  is  in 
such  a direction  that 
its  magnetic  effect 
opposes  that  of  the 
field,  as  illustrated  in 
Fig.  351,  in  which  the 
dot  or  arrow  point 
on  the  armature  con- 
ductors represents  the  current  flowing  toward  the  observer  and 
the  cross  or  arrow  feather  represents  the  current  flowing  away 
from  the  observer.  As  the  coils  move,  the  opposing  effect  of 
armature  reactions  merges  into  the  cross  effect  already  indi- 
cated, and  the  armature  reactions  therefore  tend  to  cause  a 
periodic  variation  of  the  strength  of  magnetic  flux.  When  the 
machine  under  consideration  is  operating  as  a synchronous  motor, 
the  current  under  the  poles  lagging  behind  the  impressed  voltage 
but  being  reversed  in  phase  with  respect  to  the  induced  voltage, 
evidently  tends  to  strengthen  the  fields  instead  of  to  weaken 
them. 

If  the  current  is  in  advance  of  the  phase  of  the  induced  vol- 
tage, the  armature  of  a generator  tends  to  strengthen  the  field 


Fig.  351.  — Diagram  for  showing  the  Effect  of  Armature 
Reactions  of  a Single-phase  Generator  when  the  Cur- 
rent in  the  Armature  lags  behind  the  Induced  Voltage. 


614 


ALTERNATING  CURRENTS 


magnetism  when  the  coils  are  directly  under  the  poles,  as  may 
be  understood  by  examining  Fig.  352.  On  the  other  hand,  a 
motor  armature  with  the  current  leading  the  phase  of  the  im- 
pressed voltage  tends  to  weaken  the  fields.  This  tendency  of 

the  armature  current, 
when  in  advance  of  the 
induced  voltage,  to 
strengthen  the  fields 
could  be  taken  ad- 
vantage of  to  make  an 
alternating-current 
generator  completely 
self-regulating,  or 
even  self-exciting, 
Fig.  352.  — Diagram  for  showing  the  Effect  of  Armature  through  the  action  of 
Reactions  of  a Single-phase  Generator  when  the  Cur-  jpg  armature  CUITeilt 
rent  in  the  Armature  leads  the  Induced  Voltage.  . 

lhis,  however,  would 

require  the  use  of  a condenser  or  other  source  of  capacity  react- 
ance attached  across  the  armature  terminals  to  give  the  proper 
lead  to  the  current,  which  would  not  be  practical  in  appli- 
cation. 

The  quantitative  effect  of  the  armature  current  and  the  angle 
of  lag  on  the  results  produced  by  armature  reactions  is  not  read- 
ily determined  in  single-phase  machines.  The  effect  is  periodic 
and  depends  for  its  relative  instantaneous  values  upon  the  in- 
stantaneous positions  of  the  armature  coils  with  respect  to  the 
poles,  and  the  corresponding  instantaneous  current  values.  The 
latter  depend  upon  the  effective  value  of  the  current,  the  angle 
of  lag,  and  the  form  of  current  wave.  Doubtless  the  relative 
shapes  of  armature  coils  and  pole  pieces  also  enter  the  relation. 
Moreover,  the  field  frames  are  fairly  large  masses  of  iron,  and 
they  do  not  respond  rapidly  to  changes  in  their  magnetic  sur- 
roundings ; this  apparent  magnetic  inertia  being  caused  by  the 
effect  of  eddy  currents  and  the  considerable  inductance  of  the 
field  circuit,  which  tend  to  suppress  rapid  changes  of  the  mag- 
netic flux.  It  is  therefore  reasonable  to  assume  in  general  that 
the  results  of  armature  reactions  are  less  marked  than  misfht  be 
inferred  from  a scrutiny  of  the  variation  of  the  instantaneous 
values  of  armature  magneto-motive  forces  acting  in  the  polar 
regions  of  the  magnetic  circuit. 


SYNCHRONOUS  MACHINES 


615 


The  instantaneous  value  of  the  back  turns  of  each  armature 
coil  at  any  moment  depends  upon  the  instantaneous  value  of 
the  current  multiplied  by  a function  of  the  instantaneous  position 
of  the  coil.  If  the  current  is  sinusoidal,  then  the  current  at 
any  instant  is  proportional  to  sin  a,  and  if  the  magnetizing 
effect  on  the  pole  pieces  caused  by  the  current  in  the  coil  is 
assumed  to  vary  as  a sine  function  of  the  position  of  the  coil 
with  reference  to  its  position  when  the  induced  voltage  is  zero, 
then  the  magnetizing  effect  per  ampere  is  proportional  to  cos  a. 
The  back  turns  are  then  equal  to 

V2  nl  sin(«  — 0)  cos  a 

at  each  instant,  in  which  0 is  the  angle  of  lag,  I is  the  effective 
value  of  the  current  in  the  coil,  and  n is  the  number  of  turns  in 
the  coil.  Expanding  this  expression  reduces  it  to 

7hl 

(sin  2 «cos  0 — cos  2 a sin  0 — sin  0). 

Vz 


If  0 is  zero,  this  is  a periodic  expression  of  twice  the  funda- 
mental frequency  and  of  zero  average  value.  Averaging  the 
expression  from  a = 0 to  a = 7 r,  when  0 is  not  zero,  gives  the 

value  — -^isin  0. 

V 2 

The  armature  reactions  in  polyphase  generators  are  materi- 
ally different  from  those  in  single-phase  generators,  but  consist 
essentially  of  the  sum  of  the  effects  of  the  several  phases. 
Thus,  referring  to  Fig.  350,  it  was  seen  that  when  the  current 
of  a single-phaser  is  in  phase  with  the  voltage,  magnetism  is 
crowded  into  the  trailing  pole  tips  at  each  time  of  maximum 
current,  and  resumes  its  initial  position  when  the  current  falls 
to  zero.  In  the  case  of  polyphase  machines,  in  which  the 
wire,  in  effect,  covers  the  entire  surface  of  the  armature,  the 
skewing  effects  which  the  currents  in  the  armature  coils  of 
the  several  phases  of  a balanced  machine  produce  on  the 
magnetism  of  the  field  magnet  come  into  action  succes- 
sively so  as  to  maintain  the  total  effect  uniform  when 
the  currents  are  equal  and  in  phase  with  the  induced  vol- 
tages of  their  respective  circuits.  When  the  currents  lag 


bib 


ALTERNATING  CURRENTS 


or  lead,  the  skewing  effect  decreases  and  the  back  turns 
become 


V2  nl  | sin  («  — 6)  cos  « + sin  [ a — 6 — ^-^jcos  fu  — ••• 


m ) 


. ( a 2(m  — l)7r\  ( 2(m  — l)7r' 

+ sm  « — 6 L — cos  a — 

\ m J \ m 


in  which  m is  the  number  of  phases. 

This  is  zero  and  the  back  turns  disappear  when  6 = 0,  since 


sin  «cos  a + sin 


2 7 r\  ( 2 ir\ 

it I cos  (a )+  ••• 


m 


J 


+ sin  a — 


2 ( m — 1 )7r 


m 


COS  It  — 


2 (m  — 1)7 r 


m 


= 0. 


Under  these  circumstances  the  skewing  effect  of  the  armature 
magneto-motive  force  is  a maximum. 

When  6 is  not  zero,  the  expression  becomes 

nl  r . o -of  2 77")  • 0 ( 2 (m  — 1)7 r 

sin  2 a -f  sm  2(  a - + •••  + sin 2 ( a | — 

m J 


V2l_ 


m 


9 tt\ 


cos  2 « + cos  2 ( a — - — )+  •••  + cos  2(  te- 
rn / 


2 (m  — l7r) 


m 


cos  0 
sin  6 


— m sin  6. 

But 

2 a + sin  2^a  — •••  + sin  2 — ^)  = 0, 


sin 

and 

cos 


m 


2 ct  + cos  2 (it  — + •••  + cos2(^«  — ^ = 0. 

V m J \ m J 


Therefore  the  expression  for  the  armature  magneto-motive 
force  has  the  constant  value, 

— rsin  6. 

V2 

This  is  obtained  on  the  assumptions  that  the  current  is 
sinusoidal  and  that  each  armature  coil  acts  on  the  main  magnetic 
circuit  with  an  influence  proportional  to  cos  a.  Neither  of 
these  assumptions  is  exactly  met  in  most  machines,  and  the 
back  turns  therefore  are  more  or  less  irregularly  variable 
through  each  period  of  the  armature  current.  In  case  the  cir- 


SYNCHRONOUS  MACHINES 


617 


cuit  is  unbalanced,  a pulsation  is  introduced  by  the  differences 
between  the  phases. 

The  effect  of  armature  ampere-turns  in  a single-phase  machine 
is  illustrated  in  Fig.  353,  coils  a,  a,  connected  as  in  Fig.  350, 
being  indicated.  (To  simplify  the  figure,  winding  slots  are  not 
shown.)  Curve  E is  the  induced  voltage  curve  of  the  machine 
plotted  as  a function  of  time  beginning  when  the  coils  are 
located  neutrally  in  the  magnetic  fields  as  shown  in  the  Figure, 
that  is,  at  the  instant  when  the  induced  voltage  is  zero;  and  curve 
c is  a current  curve  assumed  to  be  in  phase  with  the  induced 
voltage.  Curve  d represents  the  magnitude  of  the  magnetizing 
effect  which  would  be  produced  per  ampere  by  a continuous  cur- 
rent flowing  in  the  moving  armature  winding,  which  aids  or 
opposes  the  field  magnetization,  but  differs  in  value  with  the 
instantaneous  positions  of  the  coils,  and  is  here  plotted  with 
space  abscissas  corresponding  to  the  time  abscissas  of  curve  E. 


Fig.  353.  — Curves  showing  the  Magnetic  Activity  of  the  Armature  of  a Single-phase 
Generator  ivhen  the  Current  is  in  Phase  with  the  Induced  Voltage. 


This  magnetizing  effect  or  activity  with  a uniform  current 
flowing  in  the  armature  winding  must  evidently  have  a maxi- 
mum value  when  the  centers  of  the  coils  are  directly  under  the 
centers  of  the  poles,  i.e.  when  the  coils  are  at  the  point  shown 
in  the  figure.  When  the  centers  of  the  coils  are  ninety  elec- 
trical degrees  from  the  position  shown  in  Fig.  353,  only  a skew- 
ing effect  results  from  the  armature  ampere-turns  acting  on  the 
field  magnet,  as  is  illustrated  in  Fig.  350.  Multiplying  the 
ordinates  of  the  current  curve  with  the  corresponding  ordinates 
of  the  curve  of  magnetic  activity,  the  actual  effect  of  the  alter- 
nating current  in  the  armature  coils  may  be  obtained,  as  is 
shown  by  the  shaded  areas  oiftlined  by  curve  k.  It  is  seen  that 
the  sum  of  the  positive  and  negative  effects  is  zero,  but  that  they 
produce  the  periodical  skewing  effect  caused  by  the  armature 


618 


ALTERNATING  CURRENTS 


current  upon  the  field  magnetism  as  already  explained.  In  the 
same  manner  it  is  shown  in  Fig.  354  that  if  the  current  lags  be- 
hind or  leads  the  voltage,  alternate  loops  of  the  curve  k are  greater 
than  the  others,  and  the  field  magnets  are  periodically  either 
weakened  or  strengthened.  In  this  case,  the  effect  of  the  back 
turns  is  larger  while  the  effect  of  the  cross  turns  is  smaller. 


Fig.  354.  — Curves  showing  the  Magnetic  Activity  of  the  Armature  as  in  Fig.  353,  but 
with  the  Current  out  of  Phase  with  the  Voltage. 


Now,  in  the  case  of  a balanced  quarter-phase  alternator,  a 
second  curve  of  reaction  exactly  similar  to  k and  one-fourth 
the  pitch,  or  90  electrical  degrees,  from  k , may  be  drawn  as 
shown  in  Fig.  355  to  represent  the  action  of  the  second  set  of 
coils  when  the  current  is  in  phase  with  the  induced  voltage. 
The  vertical  lines  crossing  Figs.  353,  354,  and  355  are  located 
180  electrical  degrees  apart,  as  guides.  The  magnetizing  effect 
of  one  phase  gradually  increases  while  the  magnetizing  effect  of 
the  other  phase  decreases,  and  the  skewing  effect  becomes  nearly 
neutralized  when  the  armature  current  and  voltage  in  each 
branch  are  in  phase  with  each  other.  Similarly,  when  the  cur- 
rent lags  or  leads  in  the  two  branches  of  the  two-phase  circuit, 
the  armature  magneto-motive  forces  add  together  so  as  to  give 
a more  or  less  constant  weakening  or  strengthening  of  the  field 
magnets,  as  shown  in  Fig.  355,  where  the  ordinates  of  the  hori- 


Fig.  355.  — Curves  showing  the  Magnetic  Activity  of  the  Armature  of  a Quarter- 
phase  Generator.  Currents  in  Phase  with  Induced  Voltages. 


SYNCHRONOUS  MACHINES 


619 


zontal  line  k"  represent  this  effect.  In  Fig.  355  curves  c and 
c1  represent  the  currents  of  the  two  phases  ; d and  dx  are  the 
curves  of  activity  of  the  coils  of  the  two  phases  respectively  ; 
and  k and  kx  are  the  reaction  tendencies  of  the  separate  phases. 
The  poles  are  assumed  to  lie  midway  between  the  vertical  broken 
lines  as  in  the  previous  two  figures.  In  case  the  current  leads, 
the  fields  are  strengthened,  and  in  case  it  lags  the  fields  are 
weakened,  as  illustrated  in  Fig.  355  a.  By  the  same  process  it 
may  be  shown  that  the  armature  reaction  in  any  polyphase 
machine  is  practically  constant  when  the  system  is  balanced. 


Fig.  355  a.  — Curves  showing  the  Magnetic  Activity  of  the  Armature  as  in  Fig.  355, 
but  with  the  Currents  out  of  Phase  with  the  Induced  Voltages. 


The  effect  is  possibly  more  simply  explained  by  considering 
the  armature  resultant  magneto-motive  force  of  a polyphaser  as 
producing  a Rotating  magnetic  field,  relatively  to  the  armature 
core,  which  core  may  either  be  stationary  or  rotate.  This  is 
evidently  permissible,  as  the  armature  windings  and  currents  of 
a balanced  polyphase  armature  bear  the  same  relations  as  those 
in  the  primary  circuit  of  a rotating  field  induction  motor.  The 
creation  of  a rotating  field  is  explained  fully  in  the  chapter 
following.*  It  will  suffice  to  say  here  that  in  a polyphase  arma- 
ture the  magnetic  fluxes  of  the  phases  grow  successively  to  their 
maximum  values  in  a direction  opposite  to  the  direction  of  rota- 
tion of  the  armature.  Thus  in  a tri-phase  armature  a coil  of 
one  phase,  then  the  next  phase  to  it  against  the  direction  of  rota- 
tion, and  then  the  third  in  the  same  direction  around  the  arma- 
ture rise  to  maximum  flux,  thus  giving  an  effect  as  though  the 
flux  set  up  by  the  armature  windings  moved  360  electrical  de- 
grees with  reference  to  a point  on  the  armature  during  the 
period  of  a cycle.  Taking  this  view,  and  supposing  first  that 
the  armature  is  stationary  and  the  field  magnet  revolves,  it  is 

* Art.  185. 


620 


ALTERNATING  CURRENTS 


evident  that  a rotating  magneto-motive  force  may  be  considered 
to  be  set  up  by  the  armature  current  having  a speed  equal  to, 
or  in  synchronism  with,  that  of  the  mechanically  rotating  alter- 
nator field  magnet.  This  must  be  so  since,  when  the  alternator 
field  magnet  moves  through  360  electrical  degrees  of  rota- 
tion, it  creates  a complete  cycle  of  currents  in  the  several  arma- 
ture phases  and  hence  the  position  of  the  resultant  armature 
magneto-motive  force  for  the  coils  of  each  phase  also  moves  360 
electrical  degrees  during  the  same  period  of  time.  When  the 
power  factor  is  unity,  the  magneto-motive  forces  for  the  several 
balanced  phases  substantially  neutralize  each  other.  When  the 
armature  currents  lag,  their  combined  rotating  magneto-motive 
force  lags  in  space  with  reference  to  the  alternator  field  magnet, 
causing  a weakening  of  the  field  magnet  by  back  magnetization. 
When  the  currents  lead,  the  armature  magneto-motive  force 
steps  forward  in  its  space  phase  relation  with  the  alternator 
field  magnet,  and  causes  a strengthening  of  the  latter. 

Similar  reasoning  applies  when  the  armature  rotates  and  the 
field  magnet  is  stationary ; since,  in  that  case,  the  magneto- 
motive force  set  up  by  the  armature  currents  may  be  considered 
to  stand  in  a fixed  position  with  respect  to  the  stationary  field 
magnet  as  the  armature  revolves,  which  fixed  position  depends 
on  the  lag  of  the  currents  with  respect  to  the  induced  voltages 
in  the  branches  of  the  polyphase  windings. 

The  resultant  magneto-motive  force  always  rotates  in  rela- 
tively the  same  direction  about  the  armature  as  the  relative 
motion  of  the  alternator  field  magnet,  or  against  the  direction  of 
rotation  of  the  armature,  as  said  before.  This  may  be  proved 
by  drawing  the  curves  of  phase  currents  and  laying  out  the 
space  phase  positions  of  the  resultant  armature  magneto-motive 
force  and  alternator  magnet  for  a series  of  angular  advances  of 
the  armature  or  field. 

The  opposing  and  cross-magnetic  effects  of  lagging  armature 
currents  in  alternating  current  generators,  when  operating 
under  usual  conditions,  cause  the  external  characteristics  * to 
slope  toward  the  horizontal  axis.  This  effect  must  be  added 
to  the  slope  of  the  characteristic  caused  by  resistance  and  re- 
actance in  the  armature.  When  the  current  leads,  the  voltage 
tends  to  rise  due  to  the  strengthening  of  the  field,  a character- 

* Art.  1 55. 


SYNCHRONOUS  MACHINES 


621 


istic  that  is  made  valuable  use  of  in  maintaining  uniform  vol- 
tage at  the  receiving  ends  of  transmission  lines  through  the  use 
of  synchronous  apparatus  which  is  regulated  to  furnish  a lead- 
ing current.*  The  rise,  however,  may  be  made  excessive  by 
the  same  effect  when  the  conditions  of  the  line  and  load  are 
abnormal. 

It  is  evident  that  the  effect  of  the  armature  reaction  depends 
for  its  relative  instantaneous  values  upon  the  instantaneous 
positions  of  the  coils  with  reference  to  the  poles,  as  already 
pointed  out ; and  that  it  depends  further  for  its  actual  instan- 
taneous and  average  values  on  the  current  strength,  the  angle 
of  lag,  the  shape  of  the  current  curve,  and  the  conditions  of 
balance  in  polyphase  systems.  The  relative  shapes  of  armature 
coils  and  pole  pieces  also  enter  the  relation,  as  do  also  the  air 
space  and  tooth  reluctances,  as  well  as  the  total  reluctance  of 
the  magnetic  circuits.  Since  the  effect  of  the  reactions  is 
periodic  in  the  case  of  single  or  unbalanced  polyphase  machines, 
it  is  difficult  to  determine  accurately  its  exact  result  in  any 
particular  case,  by  any  means  except  that  of  experiment.  By 
taking  the  instantaneous  value  of  the  single-phase  current  or 
unbalanced  polyphase  current  for  a number  of  angular  positions 
of  the  armature,  the  average  skewing  and  direct  magnetizing  or 
demagnetizing  effect  of  the  armature  reaction  can,  however,  be 
approximately  determined.  The  reactions  of  polyphase  ma- 
chines, however,  may  be  determined  with  a greater  degree  of 
accuracy  and  ease  for  any  load  power  factor  when  the  sys- 
tem is  balanced,  as  has  been  shown  by  the  previously  given 
equations. 

153.  Alternator  Characteristics.  — There  are  four  curves  which 
exhibit  particularly  important  relations  between  the  functions 
of  alternators.  These  curves,  which  are  called  characteristics, 
may  be  enumerated  as  follows  : 

1.  The  Saturation  Curve  or  Curve  of  Magnetization. 

2.  The  External  Characteristic. 

3.  The  Curve  of  Synchronous  Impedance. 

4.  The  Magnetic  Distribution  Curve  and  Voltage  Curve. 

154.  Curve  of  Magnetization.  — The  curve  of  magnetization 
shows  the  relation  between  the  ampere-turns  in  the  field  wind- 
ings and  the  total  voltage  induced  in  the  armature.  From  the 

* Art.  175. 


622 


ALTERNATING  CURRENTS 


voltage  induced  in  the  armature,  the  value  of  <I>  may  be  deduced 
by  means  of  the  formula  E = ’ Provided  the  value  of  K 


is  known.*  The  value  of  K cannot  be  determined  exactly  by 
calculation,  but  may  be  ascertained  by  means  of  the  fourth 
curve  named  in  the  preceding  article.  The  experimental 
determination  of  the  curve  of  magnetization  is  carried  out 
exactly  as  in  the  case  of  direct-current  machines,  substituting 
for  the  direct-current  voltmeter  an  instrument  which  is  capable 
of  measuring  alternating  voltages.  It  is  desirable  that  the 
instrument  used  shall  indicate  the  effective  value  of  the  voltage ; 
hence,  the  measurements  should  be  made  by  a voltmeter  built 
upon  the  principles  of  either  the  hot-wire  instrument,  the 
electrostatic  instrument  modeled  after  the  quadrant  electrom- 
eter, or  the  non-inductive  form  of  high  resistance  electro- 
dynamometer, arranged  preferably  for  direct  reading  of  voltage. f 
All  instruments  used  in  alternating-current  measurements 
which  depend  for  their  indications  upon  electrodynamic  action, 
must  be  constructed  with  no  masses  of  conducting  metal  about 
them,  or  their  constants  will  depend  upon  the  frequency  of  the 
current  measured.  This  is  due  to  the  dynamic  effects  which 
eddy  currents,  circulating  in  metallic  masses,  have  on  the  cur- 
rents in  the  moving  parts  of  the  instruments.  If  a voltmeter 
has  an  appreciable  inductance,  its  reading  will  also  depend  upon 
the  frequency,  since  the  current  flowing  through  it  is  inversely 
proportional  to  VjR2  -f-  4 tt2/2!?.  From  this  it  is  readily  seen 
that  if  flu  is  not  negligible  in  comparison  with  R<  the  current 
flowing  through  the  voltmeter  when  it  is  connected  to  a circuit, 
and  hence  its  indications,  will  be  dependent  upon  the  frequency. 
The  indications  of  an  inductive  voltmeter  will  always  be 
less  when  it  is  connected  to  an  alternating-current  circuit 
than  when  it  is  connected  to  a continuous-current  circuit  of 
equal  effective  voltage.  The  self-inductance  of  an  electrody- 
namometer arranged  for  use  as  an  amperemeter  is  usually  quite 
small,  but  in  some  cases  may  reach  a millihenry.  An  electro- 
dynamometer  which  is  arranged  to  be  used  as  a voltmeter, 
usually  has  a considerable  non-inductive  resistance  in  series 
with  the  inductive  working  coils,  so  that  its  time  constant  is 
quite  small. 


* Art.  20. 


t Art.  24. 


SYNCHRONOUS  MACHINES 


623 


If  the  alternator  under  examination  is  a special  one  and  was 
designed  to  be  partially  or  entirely  a self-exciting  one,  there  is 
some  question  of  the  comparative  magnetizing  effects  of  con- 
tinuous and  rectified  currents.  In  general,  however,  the  recti- 
fied current  in  a self-excited  field  is  doubtless  always  a wavy 
one.  The  effective  value  of  this,  as  indicated  by  an  electrodyna- 
mometer, is  very  nearly  the  same  as  the  average  value  indicated 
by  an  ordinary  amperemeter.  The  average  magnetizing  effect 
of  the  current  is  also  practically  equal  to  that  of  a continuous  cur- 
rent which  gives  the  same  indications  on  the  instruments,  though 
the  actual  magnetic  flux  set  up  is  always  proportional  to  the 
instantaneous  value  of  the  current  divided  by  the  magnetic  circuit 
reluctance,  after  due  correction  has  been  made  for  hysteresis 
and  the  magnetizing  effects  of  any  eddy  currents  induced  in  the 
field  magnets  by  a pulsating  excitation.  A wavy  current  tends 
to  set  up  eddy  currents  in  the  iron  of  the  magnetic  circuit  and 
thus  cause  heating,  but  this  result  is  not  marked. 

The  exciting  current  of  a separately  excited  alternator,  the 
type  used  almost  exclusively  in  modern  practice,  may  also  be 
caused  to  become  wavy  if  the  armature  reactions  are  very  large. 
It  has  already  been  shown  that  the  effect  of  the  armature  reactions 
of  a single-phase  or  unbalanced  polyphase  machine  is  a periodic 
one  ; and  when  the  periodic  effect  becomes  of  sufficient  magni- 
tude, it  causes  fluctuations  in  the  field  magnetism,  which  react 
upon  the  windings  and  throw  the  exciting  current  into  pulsa- 
tions. On  account  of  this  tendency  of  the  exciting  current  to 
become  wavy  and  of  the  magnetic  flux  to  vary  locally  in  the  pole 
pieces,  it  is  common  to  build  the  pole  cores  of  laminations  in 
order  to  prevent  the  generation  of  excessive  eddy  currents  in 
them. 

The  general  form  of  the  curve  of  magnetization  for  an  alter- 
nator is  similar  to  the  form  of  the  curve  for  a direct-current 
dynamo.  As  it  is  not  uncommon  for  alternators  to  have  a some- 
what larger  reluctance  in  the  air  space  than  have  direct-current 
dynamos  of  the  same  size,  the  knee  in  the  alternator  curve  is 
sometimes  not  so  abrupt  as  it  is  in  the  case  of  direct-current 
machines  (Figs.  356  and  357).  For  studying  the  details  of 
the  design  of  the  magnetic  circuit,  the  curve  may  be  re- 
solved into  component  curves  representing  the  air  space, 
frame,  and  armature.  In  this  case  the  excitation  compared  to 


624 


ALTERNATING  CURRENTS 


the  magnetic  flux 
is  a straight  line 
for  the  air  space, 
since  the  reluc- 
tance of  that  part 
of  the  magnetic 
circuit  is  constant. 
Figure  357  shows 
the  relation  be- 
tween exciting 
current  and  ter- 
minal voltage  in 
a particular  alter- 
nator when  cur- 
rent flows  in  the 
armature. 

155.  External  Characteristic.  — An  external  characteristic  is  a 
curve  showing  the  relations  between  the  armature  current  and 
the  terminal  voltage  which  exist  for  a stated  condition  of  field 


Fig.  356.  - 


EXCITING  CURRENT 

- Type  of  Curve  of  Magnetization  of  an  Alternator. 


excitation.  To  ex- 
perimentally deter- 
mine an  external 
characteristic  of  an 
alternator,  it  is  ex- 
cited by  the  method 
for  which  it  is  de- 
signed, so  as  to  give 
its  normal  voltage  on 
open  circuit.  The 
volts  at  its  terminals, 
and  the  current  and 
kilowatts  in  the  ex- 
ternal circuit,  are 
measured  with  various 
loads  in  the  external 
circuit  of  various 
magnitudes  but  fixed 
power  factor.  The 
observations  may  be 
plotted  in  a curve, 


Fig.  357.  — Relation  of  Terminal  Voltage  to  Exciting 
Current  with  Different  Currents  in  the  Armature. 


SYNCHRONOUS  MACHINES 


625 


using  volts  as  ordinates  and  kilovolt-amperes  or  kilowatts  as  ab- 
scissas. In  separately  excited  alternators,  the  curve  cuts  the  ver- 
tical axis  at  its  highest  point  at  no  load,  and  then  gradually  falls, 
the  exciting  current  being  held  constant.  The  decrease  of  the 
ordinates  (drop  in  voltage)  is  caused  by  the  effects  of  armature 
resistance,  armature  self-inductance,  and  armature  reactions. 

The  magnitude  of  the  armature  reactions  is  changed  on 
account  of  the  lag  or  lead  of  the  current,  when  there  is  re- 
actance in  the  external  circuit.  If  it  is  desired  to  quanti- 
tively  determine  the  effect  of  lag  or  lead  on  armature  reactions, 
curves  may  be  taken  with  different  values  of  reactance  inserted 
in  the  external  circuit.  The  portion  of  the  drop  which  is  caused 
by  self-inductance  and  armature  reactions  when  the  machine  is 
supplying  current  to  a load  of  unity  power  factor  may  be  sepa- 
rated from  that  caused  by  resistance  by  the  formula,  E?  = Ea2 
-P  E In  this  case  is  the  open-circuit  voltage,  Ea  is  equal 
to  the  terminal  voltage  for  the  current  flowing  plus  IRa  where 
Ea  is  the  armature  resistance,  and  E,  is  the  quadrature  compo- 
nent of  the  drop  which  results  from  self-inductance  and  arma- 
ture reactions.  By  taking  the  characteristics  of  an  alternator 
when  worked  on  circuits  of  different  known  resistances  and  re- 
actances, the  effect  of  armature  reactions  may  be  determined 
for  different  values  of  the  angle  of  lag,  and  such  tests  are  quite 
valuable  for  determining  the  purpose  for  which  the  machine  is 
best  adapted.  The  self-inductance  and  reactions  of  some  of 
the  early  alternators  were  so  great  that  the  external  character- 
istic drooped  to  the  X axis  at  a current  not  greatly  exceeding 
full  load.  Figure  358  shows  the  characteristics  of  two  alter- 
nators plotted  to  show  the  relations  of  terminal  volts  to  arma- 
ture current  when  the  load  is  non-reactive.  One  (Curve  A')  is 
of  a standard  type  of  alternator,  having  small  armature  self- 
inductance and  reactions,  and  the  second  (Curve  B ) of  a special 
machine  having  large  armature  self-inductance  and  reactions. 
Curves  showing  the  relation  of  output  and  armature  current 
for  the  two  machines  are  also  shown.  These  are  especially 
enlightening  as  showing  relatively  how  much  lower  the  output 
of  the  highly  inductive  machine  is  for  the  same  proportions  of 
full  load  current.  The  two  curves  of  machine  B illustrate  the 
difficulty  of  obtaining  good  regulation  or  the  economical  utili- 
zation of  the  copper  and  iron  in  the  machine  when  the  machine 


620 


ALTERNATING  CURRENTS 


is  highly  reactive.  As  inductive  reactance  in  the  load  has  the 
same  effect  upon  these  characteristics  of  operation  as  has  react- 
ance in  the  armature  itself,  it  is  evident  from  the  figure  that 
inductive  reactance  should  be  avoided  in  both  the  machine  itself 


0 50  100  150  200 


ARMATURE  CURRENT  IN  PER  CENT  OF  FULL  LOAD  CURRENT 

Fig.  358. — External  Characteristics.  Machines  worked  on  Non-reactive  Loads. 
Machine  A with  Small  and  Machine  li  with  Large  Armature  Reactance.  The  Rela- 
tions of  Output  to  Current  are  also  shown. 


and  its  load  so  far  as  possible.  The  three  following  figures 
(Figs.  359,  360,  and  361)  graphically  indicate  the  effect  of  an 
inductive  load.  Figure  359  shows  the  relation  between  voltage 
drop  and  load  power  factor  upon  a generator  of  normal  design 
connected  to  inductive  loads  of  different  power  factors.  In 
this  case,  the  load  was  at  the  end  of  a transmission  line.  The 
drops  in  the  step-down  transformers,  line,  and  generator,  and  their 
total  are  given  for  power  factors  ranging  from  100  per  cent  to 
20  per  cent,  inductive,  when  the  full  load  rating  of  250  kilovolt- 
amperes are  being  delivered  to  the  transmission  line.  Figure 
360  shows  the  reduction  in  the  kilowatts  of  capacity  available 
in  this  particular  generator  when  used  on  the  same  range  of 
power  factors  and  with  a kilovolt-ampere  output  for  the  various 


SYNCHRONOUS  MACHINES 


627 


Fig.  359.  — Curves  showing  the  Effects  of  the  Power  Factor  of  Inductive  Loads  or 
the  Regulation  of  an  Alternator. 


Fig.  360.  — Curves  showing  the  Relation  of  Inductive  Power  Factor  to  Kilowatts  and 
Kilovolt-amperes  for  the  Alternator  referred  to  in  Fig.  359. 


628 


ALTERNATING  CURRENTS 


power  factors  as  shown  by  the  upper  curve.  The  lessening  of 
the  capacity  is  clue  to  the  fact  that  the  output  of  the  generator 
is  limited  by  the  heating  of  the  armature,  and  therefore  when 
the  current  contains  a large  quadrature  component,  the  active 
component  must  be  relatively  low.  Figure  361  shows  the  field 
exciting  currents  required  by  the  alternator  to  supply  normal 
voltage  to  the  transmission  line  at  the  same  range  of  power 
factors,  when  the  alternator  is  operated  so  as  to  furnish  250 
kilovolt-amperes,  and  it  also  shows  the  corresponding  armature 
core  losses. 


Eig.  361. — Curves  showing  the  Variation  of  Field  Exciting  Current  and  Armature 
Core  Losses  with  the  Power  Factor  of  an  Inductive  Load  for  the  Alternator  referred 
to  in  Figs.  359  and  360. 


When  the  load  has  the  effect  of  electrostatic  capacity  the 
quadrature  leading  current  tends  to  strengthen  the  alternator 
field  magnet  and  neutralize  the  effect  of  armature  self-induct- 
ance. It  may,  therefore,  reduce  or  neutralize  the  drop  in  the 
external  characteristic,  or  cause  it  to  rise,  depending  upon  the 
amount  of  the  quadrature  current  flowing. 

The  external  characteristic  of  a special,  self-excited,  shunt- 
wound  alternator  is  shown  in  Fig.  362.  The  droop  in  the  curve  is 
a little  greater  than  would  occur  for  the  same  machine  separately 


SYNCHRONOUS  MACHINES 


629 


“2000 


£1000 


excited.  This  is  due  to  3000 
the  loss  of  exciting  current 
as  the  armature  drop  in- 
creases, and  the  terminal 
voltage  decreases  accord- 
ingly- 

The  effect  of  armature 
resistance  and  true  self- 
inductance can  be  shown 

by  drawing  the  loci  of  the  pIG  352.  — External  Characteristic  of  a Special 
alternator  terminal  voltage  Self-excited  Alternator. 

— maintained  constant  by  proper  variation  of  the  field  excita- 
tion — when  the  power  factor  of  the  load  is  varied,  but  the 
load  current  is  maintained  constant.  The  construction  is  shown 
in  Fig.  363. 


-■£2*2*4 1 

10 


13 


ARMATURE  AMPERE8 


In  this  figure  the  circular  arc  EExE^E^Ei  with  the  center  at 
0 is  the  locus  of  constant  terminal  voltage  when  the  constant  cur- 
rent 01  flows  but  the  power  factor  is  varied.  The  triangle  OXR 
represents  the  voltage  drops  in  the  impedance  of  the  armature 
by  reason  of  the  flow  of  current  OL  Then  the  total  voltage 
generated,  E',  must  be  equal  to  the  vector  sum  of  OX  and  the 
terminal  voltage  E.  By  geometrical  construction  it  is  seen 
that  the  locus  of  E\  = OE',  etc.)  for  various  angles  of  lag  6 
in  the  external  circuit  must  be  the  circular  arc  E'  EXE^E^EX 
with  its  center  at  X.  I11  the  figure,  when  01  lags  by  the 
angle  6 with  respect  to  the  terminal  voltage,  the  latter  is 
represented  by  OE , the  total  voltage  generated  by  OE ’,  and 
the  impedance  drop  by  EE',  of  which  Ea  is  used  in  overcoming 
self-inductance  and  aE'  in  overcoming  resistance.  The  figure 
shows  the  vector  diagrams  for  lagging  angles  6 and  0V  zero  lag  $2, 
and  leading  angles  03  and  0X,  where  the  thetas  represent  the 
phase  differences  of  the  current  and  voltage  in  the  external 
circuit.  It  will  be  noted,  in  order  that  constant  current  may 
flow  with  varying  power  factors,  the  load  must  be  assumed  to 
change  in  its  components  of  resistance  and  self-inductance  or 
capacity  in  such  a way  as  to  keep  the  impedance  Z constant. 
The  line  EX  in  the  figure  is  assumed  to  be  constant,  which 
is  not  absolutely  true,  as  the  reluctance  of  the  self-induction 
paths  around  the  armature  conductors  will  change  with  varying 
values  of  theta.  In  this  figure  the  values  of  total  voltage  in- 


630 


ALTERNATING  CURRENTS 


duced,  OE' , OE^,  etc.,  are  due  to  cutting  the  total  flux  in  the 
air  gap,  which  in  turn  is  created  by  the  vector  sum  of  the  field 
ampere-turns  and  effective  armature  ampere-turns.  The  latter 


Fig.  363.  — Loci  of  Terminal  and  Total  Voltages  generated  in  an  Alternator  when 
the  Power  Factor  of  the  Load  is  varied  but  the  Numerical  Values  of  Terminal 
Voltage  and  Current  are  maintained  constant. 

vary  with  the  positions  of  the  centers  of  the  armature  coils  at 
the  instant  they  are  carrying  maximum  current,  as  shown  in  an 
earlier  article,  or  approximately  with  sin  0.*  This  shows  that 

* Art.  152. 


SYNCHRONOUS  MACHINES 


631 


when  9 is  positive  the  field  ampere-turns  must  not  only  create 
sufficient  flux  to  create  the  total  voltage  required,  but  must  also 
neutralize  the  effective  armature-turns,  but  that  when  9 is 
negative  the  field  ampere-turns  directly  add  to  the  effective 
armature  ampere-turns  to  set  up  the  required  flux.  A diagram 
which  expresses  graphically  the  effect  of  armature-turns  upon  the 
voltage  generated 
is  shown  in  Fig. 

364.  T r ian  g 1 e 
OFC  represents 
the  voltage  triangle 
of  the  external  cir- 
cuit, when  the  gen- 
erator terminal 
voltage  is  OC,  the 
angle  of  lag  9,  and 
the  current  01. 

Triangle  CD  A rep- 
resents the  internal 
drops  in  the  ma- 
chine due  to  self- 
inductance and  re- 
sistance, the  total 
drop  being  CA. 

OA  is  the  voltage  that  would  be  required  to  drive  the  cur- 
rent 01  through  an  impedance  equal  to  This  voltage, 

however,  is  less  than  would  be  generated  by  the  same  field 
excitation  on  open  circuit  by  reason  of  the  demagnetization 
caused  by  the  current  flowing  in  the  armature  coils.  Repre- 
senting the  loss  of  voltage  from  this  source  by  AB,  then  the 
open  circuit  voltage  is  represented  by  OB.  Or,  a field  excita- 
tion which  would  create  an  armature  voltage  OB  is  required  to 
furnish  a terminal  voltage  OC  when  current  01  flows  at  an 
angle  of  lag  9 behind  OC.  The  triangle  OAB  may  then  be 
considered  proportional  to  the  ampere-turns  of  the  machine,  in 
which  OB  is  proportional  to  the  useful  field  ampere-turns.  AB 
the  armature  ampere-turns,  and  OA  the  net  ampere-turns  effec- 
tive in  producing  useful  flux  in  the  armature.  It  will  be  noted 
that  the  total  field  ampere-turns  must  include  the  ampere-turns 


Fig.  364.  — Vector  Diagram  showing  the  Effect  of  Ar- 
mature Magnetizing  Ampere-turns  upon  the  Voltage 
generated  in  an  Alternator.  Armature  Current  lagging 
behind  Terminal  Voltage. 


632 


ALTERNATING  CURRENTS 


which  set  up  leakage  flux  ; and  these  vary,  since  the  leakage 
increases  with  a positive  lag  of  the  current  and  decreases  with 

a negative  lag. 
Since  OB  does  not 
include  ampere- 
turns  necessary  to 
set  up  leakage  flux, 
the  figure  is  onty 
approximate.  Fig- 
ure 365  is  a diagram 
showing-  the  effect 


C D 

Fig.  365.  — Diagram  showing  the  Effect  of  Armature 
Magnetizing  Ampere-turns  upon  the  Voltage  Gener- 
ated iu  an  Alternator.  Armature  Current  leading  the 
Terminal  Voltage. 


of  armature  magnetization  when  the  current  in  the  external 


circuit  leads  the  terminal  voltage. 


156.  The  Short-Circuit  Current  Curve  and  Synchronous  Imped- 
ance of  an  Alternator.  — From  the  discussion  preceding  it  is 
evident  that  there  is  difficulty  in  separating  the  effects  of  true 
self-inductance  and  armature  reactions  in  an  alternator,  but 
the  combined  effect  of  the  two  in  connection  with  the  armature 
resistance  may  be  studied  to  advantage  by  the  use  of  the  Short- 
circuit  current  curve.  This  curve  shows  the  relation  between 
the  ampere-turns  of  the  field  magnet  and  the  armature  current 
flowing  when  the  armature  terminals  are  short-circuited.  The 
curve  may  be  obtained  by  placing  an  amperemeter  in  the  field 
circuit  and  short-circuiting  the  armature  directly  across  its  own 
terminals  except  that  a very  low  resistance  amperemeter  should 
be  placed  in  at  least  one  of  the  leads.  The  field  magnet  may  be 
then  excited  by  successively  increasing  values  of  exciting  cur- 
rent and  the  readings  of  the  amperemeters  observed  and  plotted, 
this  process  being  continued  until  the  maximum  safe  value  of  the 
armature  current  is  reached.  The  machine  should  be  driven 
at  normal  speed.  As  in  the  case  of  the  saturation  curve,  the 
curves  of  rising  and  falling  field  excitation  will  differ  on  account 
of  the  effect  of  hysteresis  in  the  field  magnet.  In  order  to  ob- 
tain all  the  information  desirable,  a saturation  curve  for  the 
same  excitations  should  also  be  obtained  and  should  be  plotted 
on  the  chart  with  the  short-circuit  current  curve.  Figure  366 
shows  a short-circuit  curve  of  an  1800-kilowatt,  tri-phase  ma- 
chine, marked  A.  Curve  B is  the  saturation  curve  of  the 
same  machine,  and  the  points  R and  S are  the  points  for  full 
load  rated  voltage  and  current  respectively. 


TERMINAL  VOLTS 


SYNCHRONOUS  MACHINES 


633 


From  these  two  curves  an  imaginary  impedance  may  be 
obtained  corresponding  with  each  armature  current,  which  is 
approximately  equal  to  an  actual  impedance  calling  for  the 
same  loss  of  voltage  as  is  jointly  caused  by  the  armature  resist- 
ance, armature  self-inductance,  and  armature  reactions.  This 
is  called  the  Synchronous  impedance,  and  is  sometimes  defined 
as  the  numerical  ratio  of  the  open-circuit  voltage  generated  at 
normal  speed  for  a given  field  excitation  to  the  short-circuit  cur- 
rent at  the  same  speed  and  excitation.  The  term  synchronous 
impedance,  however,  can  be  better  considered  as  designating  the 


7000 

6000 

5000 

4000 

3000 

2000 

1000 

0 


C/ 

B 

^ / 

r 

J 

r / 

\ X 

V 

/ 

✓ 

✓ 

/ 

/ 

' 

/ / 
/ / 

x / 

/ | 

/ 

/ ' 
✓ 1 
✓ . 

/ 

A 

/ / 
/ / 

V' 

/ 

/ 

/ 

A 

v >.✓ 

/ 

/ 

/ ^00^ 

K| 

0 N M2000  4000  6000  8000  10000  12000  14000  16000  18000 

AMPERE  TURNS  ON  EACH  FIELD  COIL 


Fig.  366.  — Short-circuit  Current  Curve  of  an  Alternator.  A,  Short-circuit  Curve; 


B,  Magnetization  Curve. 


actual  vector  loss  of  voltage,  due  to  armature  resistance,  self- 
induction,  and  reactions,  divided  by  the  armature  current  flow- 
ing. Its  value,  therefore,  changes  with  the  angle  of  lag  of  the 
armature  current  on  account  of  the  resultant  change  in  effective 
armature  magnetizing  turns. 

A curve  which  is  useful  in  determining  the  excitation 
required  to  give  a desired  terminal  voltage  or  vice  versa , when 
a given  current  flows  at  a given  angle  of  lag,  is  marked  MGrL 
in  Fig.  366  and  is  sometimes  called  the  wattless  current  char- 
acteristic. The  curve  is  constructed  in  this  way:  assume  some 


ARMATURE  AMPERES  PER  PHASE 


634 


ALTERNATING  CURRENTS 


armature  current,  say  50  amperes,  and  lay  off  the  distance  OM 
equal  to  the  number  of  ampere-turns  per  pole  that  are  shown  by 
the  short-circuit  curve  to  be  necessary  to  cause  a short-circuit 
current  of  the  chosen  value  to  flow  through  the  armature  cir- 
cuit. Then  lay  off  MN,  which  is  the  back-magnetizing  turns 
per  pole  caused  by  the  armature  current  of  50  amperes  when 
flowing  at  an  inductive  lag  angle  of  90°  and  is  computed  for  a 
polyphase  machine  by  the  formula  on  page  616  and  for  a single- 
phase machine  by  the  formula  on  page  615.  To  do  this  the  ar- 
mature resistance  must  be  considered  negligible,  which  for 
ordinary  machines  is  admissible.  Then  ON  represents  the  am- 
pere-turns per  pole  in  the  field  windings  which  are  effective  in 
setting  up  useful  flux,  and  I)N  (which  is  the  induced  voltage 
corresponding  to  field  ampere-turns  ON)  represents  the  induc- 
tive drop  in  the  armature.  The  field  reaction  of  a constant 
quadrature-lagging  current  in  the  armature,  and  the  self-induct- 
ance set  up  thereby,  are  constant  whatever  the  terminal  voltage 
may  be,  provided  the  effect  of  saturation  may  be  neglected,  and 
therefore  if  the  machine  is  loaded  with  pure  inductive  reactance 
and  the  field  current  modified  so  as  to  keep  the  armature  cur- 
rent constant,  the  relation  between  terminal  voltage  and  field 
current  will  be  expressed  by  curve  MOL  of  Fig.  366,  the  offsets 
HGr,  PL , etc.,  being  all  parallel  and  equal  to  PM.  Then,  if 
the  reactive  load  is  such  that  the  terminal  voltage  is  OK  and 
the  ampere-turns  per  field  spool  are  ON  Gr  will  be  one  point  on 
the  wattless  current  characteristic  and  GrH  must  equal  and  be 
parallel  to  PM.  The  curve  MGrL  is  evidently  constructed  by 
drawing  it  through  the  extremities  of  lines  drawn  from  curve 
B equal  and  parallel  to  PM.  For  greater  currents  MGL  is 
further  removed  from  B. 

The  line  00  represents  approximately  the  ampere-turns  re- 
quired by  the  air  space. 

157.  The  Magnetic  Flux  Distribution  Curve  of  an  Alternator, 
and  Curves  of  Voltage  and  Current.  — These  curves  may  be  ex- 
perimentally determined  by  various  methods.  They  consist  of  a 
series  of  curves  which  are  closely  interrelated,  but  may  be,  and  in 
fact  are  likely  to  be,  of  quite  dissimilar  forms.  The  form  of  the 
curve  representing  the  wave  of  impressed  voltage,  or  total  vol- 
tage induced  in  the  armature  of  an  alternator,  is  dependent  upon 
the  distribution  of  the  magnetic  flux  over  the  pole  faces,  and  also 


SYNCHRONOUS  MACHINES 


635 


on  the  arrangement  of  the  armature  windings.  By  design,  either 
of  these  may  be  given  a controlling  influence  to  the  exclusion  of 
the  other.  A two-pole  machine  with  closed-circuit,  evenly  dis- 
tributed, Gramme  ring  armature  winding,  as  shown  in  Fig.  30, 
gives  an  excellent  illustration  of  this,  when  it  has  two  connections 
180  degrees  apart  to  two  collector  rings,  thus  connecting  the 
two  halves  of  the  winding  in  parallel  between  the  rings.  The 
differential  action,  which  occurs  in  the  coils  of  such  an  arma- 
ture, makes  its  curves  of  voltage  almost  independent  of  the 
distribution  of  the  magnetism  over  the  pole  faces,  provided 
the  same  total  number  of  lines  of  force  is  cut  by  each  con- 
ductor per  revolution;  and  the  maximum  value  of  the  voltage 
is  entirely  independent  of  the  magnetic  flux  distribution. 
This  is  shown  by  Fig.  367,  where  the  full  line  in  the  left- 
hand  drawing  shows  the  curve  of  voltage  developed  in  such 


Fig.  367. — Voltage  Curves  produced  by  a Two-pole,  Gramme  Ring  Armature 
Winding  having  Two  Coils  in  Parallel  and  each  Coil  covering  180  Electrical  De- 
grees, when  subjected  to  Magnetic  Fluxes  of  Differing  Distribution.  Full  Lines 
represent  Voltages  and  Dotted  Lines  Magnetic  Distribution  under  the  Pole  Faces. 

an  armature  winding  with  a uniform  distribution  of  the 
magnetic  flux  of  the  field,  and  the  full  line  in  the  other  draw- 
ing shows  the  curve  of  voltage  with  the  same  total  field  greatly 
distorted.  The  dotted  lines  in  the  drawing  indicate  the  dis- 
tributions of  the  magnetism  over  the  pole  faces.  The  unidirec- 
tional voltage  delivered  by  the  armature  when  the  windings 
are  connected  to  a commutator  and  the  machine  is  operated 
as  a direct-current  dynamo,  is  not  affected  by  the  distortion, 
provided  the  brushes  are  always  placed  on  the  neutral  plane 
and  the  total  magnetism  passing  through  the  armature  re- 
mains constant.  When  the  machine  has  been  converted  into 
an  alternator  by  the  addition  of  collector  rings  as  stated,  the 
maximum  instantaneous  voltage  is  equal  to  the  voltage  devel- 
oped in  the  direct-current  machine,  and  is  therefore  independ- 


636 


ALTERNATING  CURRENTS 


ent  of  the  distribution  of  the  magnetism,*  and  the  form  of  the 
curve  representing  the  wave  of  voltage  is  but  slightly  altered, 
as  shown  in  Fig.  367.  Now  suppose  the  same  machine  to  be 
arranged  with  a single  narrow  coil  on  the  armature.  The 
change  in  the  magnetic  distribution  now  not  only  changes 
the  form  of  the  voltage  curve  proportionally,  but  also  changes 
in  a marked  manner  the  maximum  value  of  the  instantaneous 
voltage.  The  difference  between  the  curves  of  voltage  devel- 
oped by  the  broad  coil  and  narrow  coil  armatures  is  due  to 
the  effect  of  differential  action  in  the  broad  coil  armature. 

It  has  already  been  shown  that  differential  action  occurs 
to  some  degree  in  commercial  alternators,  but  it  does  not 
occur  to  a sufficient  degree  to  make  the  form  of  the  voltage 
wave  independent  of  the  magnetic  distribution.  It  is  there- 
fore true  that  the  magnetic  distribution  largely  influences 
the  form  of  the  voltage  wave,  and  the  magnetic  distribution 
should  therefore  be  carefully  studied  during  the  development 
of  a type  of  alternators.  A proper  study  of  the  magnetic 
flux  distribution  and  of  the  arrangement  of  the  armature 
windings  makes  it  possible  to  so  design  an  alternator  that 
it  will  produce  any  desired  form  of  voltage  wave. 

The  angular  relation  between  the  curves  representing  the 
magnetic  flux  distribution  and  the  impressed  voltage  is  inter- 
esting. The  ordinate  of  the  curve  of  voltage  at  any  instant  is 
proportional  to  the  rate  at  which  lines  of  flux  are  cut  by  the 
armature  conductors  at  that  instant,  the  rate  being  taken  alge- 
braically. Consequently,  the  voltage  is  zero  when  the  mag- 
netization is  all  symmetrically  threaded  through  the  coils;  that 
is,  when  the  algebraic  net  rate  of  cutting  flux  by  the  conductors 
is  zero.  The  voltage  is  a maximum  when  the  rate  of  cutting 
flux  is  the  greatest ; that  is,  when  the  algebraic  summation  of  the 
number  of  flux  lines  threaded  through  the  coils  is  a minimum.  A 
curve  which  represents  the  algebraic  summation  of  the  number 
of  flux  lines  threaded  through  the  coils  at  each  instant,  therefore, 
has  an  angular  position  which  is  90°  from  that  of  the  voltage 
curve  (Fig.  368).  The  form  and  dimensions  of  this  curve  of 
magnetic  linkages  evidently  depend  upon  the  actual  distribu- 
tion of  the  magnetic  flux  and  the  arrangement  of  the  armature 
windings.  When  the  voltage  curve  is  irregular,  and  the  angular 

* Art.  20. 


SYNCHRONOUS  MACHINES 


637 


relation  is  not  made  evident  from  the  curve,  as  in  Fig.  369, 
the  point  at  which  the  curve  of  magnetic  linkages  cuts  the 
X-axis  may  yet  be  easily  found, 
since  it  is  directly  under  the  center 
of  gravity  of  the  voltage  curve. 

That  is,  since  as  manjr  lines  of  flux 
must  be  withdrawn  from  the  coils 
as  are  inserted,  for  each  loop  of  the 
curve,  the  summation  of  the  ordi- 
nates of  the  voltage  curve  on  each 
side  of  the  crossing  must  be  equal. 

This  curve,  which  shows  the  alge- 
braic number  of  magnetic  lines  of  flux 
which  are  threaded  at  each  instant 
through  the  coils,  may  be  deduced  from  a known  curve  of  voltage. 
Erect  an  ordinate  to  the  voltage  curve  which  bisects  the  area  as 

in  Fig.  369 ; then 
by  means  of  ordi- 
nates divide  the  half 
areas  into  a number 
of  small  areas.  The 
magnetism  threaded 
through  the  armature 
coils  is  algebraically 
equal  to  zero  at  the 
instant  represented 
by  the  bisecting  or- 
dinate, and  the  al- 
gebraic value  of  the 
magnetic  leakages 
through  the  armature 


Fig.  368. — Curves  showing  Relation 
of  Magnetic  Flux  Linkages  and 
Induced  Voltage  in  an  Alternator. 


Fig.  369.  — Alternator  Voltage  and  Magnetic  Linkage  coils  at  anv  other  ill- 
Curves.  Area  of  Voltage  Curve  is  bisected.  J . 

stant  is  proportional 

to  the  area  inclosed  by  the  voltage  curve  between  the  corre- 
sponding ordinate  and  the  bisecting  ordinate,  since  e = ~ and 

do 


$ = "Zedt,  where  e is  instantaneous  voltage,  <fi  magnetic  flux 
linkages,  and  t time.  Therefore,  the  ordinates  of  the  curve 
representing  the  magnetic  linkages  through  the  coils  are,  at 
the  instants  represented  by  the  ordinates  which  divide  the 


638 


ALTERNATING  CURRENTS 


voltage  curve  into  small  areas,  proportional  to  the  area  be- 
tween the  corresponding  instantaneous  voltage  ordinate  and  the 
bisecting  ordinate.  The  full  process  is,  therefore,  as  follows : 
lay  off  on  each  ordinate  a length  from  the  X-axis  proportional 
to  the  area  inclosed  by  the  voltage  curve  between  the  respec- 
tive ordinate  and  the  bisecting  ordinate.  The  points  thus 
found  are  points  on  the  desired  curve.  It  is  evident  that  the 
maximum  ordinate  of  the  curve  comes  at  the  instant  when  E 
is  equal  to  zero.  The  length  of  this  ordinate  is  equal  to  <J>,  the 
maximum  value  of  the  magnetic  flux  linkages,  and  the  scale  of 
the  curve  may  thus  be  conveniently  fixed.  The  loops  may  not 
be  symmetrical,  but,  with  a fixed  value  of  <E>  and  a fixed  arma- 
ture winding,  the  successive  loops  must  always  be  exactly  alike, 
though  they  may  be  looked  upon  as  alternately  positive  and  neg- 
ative, since  the  magnetic  flux  is  alternately  threaded  through 
the  coils  in  opposite  directions. 

The  corresponding  curves  of  voltage  and  magnetic  linkages 
for  various  forms  of  voltage  curves  are  shown  in  the  accompany- 
ing figures.  In  Fig.  369  the  voltage  curve  is  one  experimen- 
tally determined  from  a special  alternator  when  working  on  a 
series  load  of  arc  lights.*  In  Fig.  370  the  voltage  curve  is  an 


Fig.  370.  — Curve  e is  a Triangular 
Voltage  Curve ; Curve  m,  Magnetic 
Flux  Linkages. 


Fig.  371.  — Curve  e,  Sinusoidal  Voltage 
Curve;  Curve  in,  Magnetic  Flux 
Linkages. 


isosceles  triangle,  in  Fig.  371  it  is  a sinusoid,  and  in  Fig.  372  it 
is  a rectangle.  The  voltage  curves  of  Figs.  373  and  374  are 
respectively  a flat-topped  curve  and  a parabola. f 

Since  the  induced  voltage  is  proportional  to  the  rate  of 
change  of  the  number  of  lines  of  flux  threaded  through  the 

* Tobey  and  Walbridge,  Stanley  Alternate-Current  Arc  Dynamo,  Trans. 
Amer.  Inst.  E.  E.,  Vol.  7,  p.  367. 

f Emery,  Alternating  Current  Curves,  Trans.  Amer.  Inst.  E.  E.,  Vol.  12, 
p.  433. 


SYNCHRONOUS  MACHINES 


639 


Fig.  372. — Curve  e,  Rectangular  Voltage 
Curve ; Curve  m,  Magnetic  Flux  Link- 
ages. 


Fig.  373.  — Curve  e,  Flat-topped  Vol- 
tage Curve ; Curve  in,  Magnetic  Flux 
Linkages. 


armature  coils,  the  ordinates  of  the  voltage  curve  are  pro- 
portional to  the  tangents  of  the  curve  representing  the  number 
of  magnetic  linkages  through  the  armature  coils.  Figure  375 
shows  a graphical  construction  for  determining  the  voltage 
curve  from  this  magnetic  flux 
linkage  curve.  Ordinates  Oa' , 

Ob' , and  OA  are,  by  construc- 
tion, made  proportional  in 
length  to  the  tangents  of  the 
angles  made  with  the  X-axis 
by  lines  tangent  to  the  mag- 
netic curve  at  pv  p2,  and  0. 

The  point  O'  is  a point  taken 
at  any  convenient  position,  and  the  lines  O'  a' , O'b ',  O'  A are 
drawn  parallel  respectively  to  aa , bb,  and  the  tangent  to  the 


Fig.  374. — Cur  ve  e,  Parabolic  Voltage  Curve ; 
Curve  in,  Magnetic  Flux  Linkages. 


Fig.  375. — Construction  showing  Method  of  Determining  Voltage  Curve  from  th« 
Magnetic  Flux  Linkage  Curve. 


640 


ALTERNATING  CURRENTS 


magnetic  curve  at  0.  The  points  of  intersection  of  horizontal 
lines  drawn  from  a b' , etc.,  and  vertical  lines  drawn  from  pv 
pv  etc.,  are  points  y>2',  etc.,  which  are  located  on  the  re- 
quired voltage  curve. 

Another  directly  useful  magnetic  flux  distribution  curve  is 
one  showing  the  distribution  of  the  lines  of  flux  over  the  pole 
faces.  This  curve  is  analogous  to  the  magnetic  flux  distri- 
bution curve  of  direct-current  machines.  It  may  be  experimen- 
tally determined  by  using  a narrow  test  coil  connected  to  a 
ballistic  galvanometer  and  fastened  to  the  armature  surface.  By 
rotating  the  field  magnet  or  armature  by  equal  small  increments 
through  360  electrical  degrees,  the  galvanometer  readings 
plotted  in  the  form  of  a curve  will  have  ordinates  proportional 
to  those  of  the  curve  of  magnetic  flux  distribution.  If  the 
number  of  turns  of  the  test  coil  and  the  constants  of  the  gal- 

O 

vanometer  are  known,  the  number  of  flux  lines  cut  by  the  coil 
are  at  once  determinable.  When  this  method  is  used,  it  is 
convenient  to  wrap  the  test  coil  across  the  face  of  the  armature 
and  around  the  core  like  a ring  winding  when  the  armature 
construction  permits  this  arrangement.  The  number  of  turns 
needed  in  the  test  coil  depends  upon  the  density  of  the  mag- 
netic flux  and  the  constants  of  the  galvanometer.  It  can  readily 
be  determined  by  trial.  Another  method  is  to  make  a narrow 
test  coil  of  pancake  shape  as  long  as  the  armature  coils.  This 
is  laid  upon  the  armature  surface.  The  field  current  is  then 
reversed  and  the  galvanometer  throw  read.  The  field  magnet 
or  armature  is  then  rotated  a distance  equal  to  the  width  of  the 
coil  and  another  reading  taken,  and  so  on  through  360  electrical 
degrees.  The  sides  of  the  coil  in  this  case  should  be  very  nar- 
row so  that  all  of  the  turns  will  inclose  essentially  all  of  the 
magnetic  flux  passing  within  the  boundary  of  the  coil. 

If  the  average  distribution  of  magnetic  flux  over  the  pole 
faces  of  a machine  is  known,  it  is  evidently  possible  to  approxi- 
mately determine  the  form  of  the  voltage  curve  which  will  be 
produced  b}r  any  particular  arrangement  of  the  windings.  It  is 
also  equally  possible  to  determine  the  arrangement  of  the  wind- 
ing required  to  give  any  desired  form  of  voltage  curve.  Again, 
if  a particular  form  of  winding  is  desirable,  the  magnetic  flux 
distribution  which  is  necessary  to  give  a desired  voltage  curve 
may  be  determined.  This  distribution  may  then  be  used  as  a 


SYNCHRONOUS  MACHINES 


641 


guide  in  designing  the  width  and  shape  of  the  pole  faces.  The 
application  of  the  magnetic  flux  distribution  curve  is  illus- 
trated in  Fig.  376.  The  dimensions  and  form  of  the  pole  pieces 
and  of  an  armature  coil  belonging  to  an  alternator  are  indi- 
cated in  the  figure.  The  ordinates  of  the  line  ABCD  repre- 
sent the  magnetic  density  in  the  air  space.  When  the  coil  is  in 
the  instantaneous  position  represented,  the  value  of  the  voltage 
is  zero.  As  the  coil  moves,  each  conductor  cuts  lines  of  flux. 
Suppose  that  in  a frac- 
tion such  as  a twelfth  of  a 
period  the  coil  has  moved 
from  the  position  indi- 
cated by  the  letters  x,  y , 
to  x ',  y' . During  this 
motion  each  conductor 
has  cut  a certain  number 
of  lines  of  force,  and  the  Fig.  370. — Diagram  indicating  Use  of  Magnetic 
number  cut  by  all  the  con-  Flux  Distribution  Curve  for  determining  Form 
, . , of  Voltage  Curve, 

ductors  is  approximately 

proportional  to  the  sum  of  the  areas  of  the  curve  ABCD  taken 
from  x to  x'  and  from  y to  y' . The  shorter  the  step  taken,  the 
more  accurate  this  becomes.  The  average  voltage  developed 
during  this  interval  is  also  proportional  to  the  same  area. 
Consequently  an  ordinate  which  is  numerically  equal  to  the 
area  may  be  erected  at  the  point  corresponding  to  the  inter- 
mediate position  of  the  coil  during  the  increment  motion,  to 
approximately  represent  the  voltage  at  that  point  in  the  revo- 
lution of  the  armature.  This  proceeding  may  be  repeated  step 
by  step  through  the  half  period,  taking  the  algebraic  summation 
of  the  voltage  developed  in  the  two  halves  of  the  coil,  and 
the  outline  of  the  voltage  curve  which  corresponds  to  the  par- 
ticular arrangement  of  armature  conductors  and  flux  distri- 
bution is  thus  determined. 

The  curve  representing  the  current  wave  also  represents, 
when  taken  to  the  proper  scale,  the  curve  of  active  voltage. 
From  preceding  pages  it  is  evident  that  the  curves  of  active 
and  impressed  voltages  have  the  same  forms  and  are  superposed, 
if  the  circuit  in  which  they  act  is  non-reactive.  They  have  the 
same  form,  also,  when  the  circuit  is  reactive  if  the  impressed 
voltage  is  sinusoidal,  provided  the  inductance  or  capacity 


642 


ALTERNATING  CURRENTS 


causing  the  reactance  is  independent  of  the  instantaneous  values 
of  the  voltage  and  the  current  and  the  armature  reactions  are 
approximately  uniform.  The  condition  of  a uniform  inductance 
can  only  hold  when  no  iron  is  inclosed  in  any  portion  of  the  circuit. 
In  general  this  condition  is  not  found  in  commercial  service. 
Even  with  a uniform  inductance  the  curves  of  impressed  and 
active  voltages  will  not  coincide  in  phase,  since  phase  coinci- 
dence between  them  can  occur  only  when  the  circuit  is  without 
either  inductance  or  capacity,  or  these  exactly  neutralize  each 
other.  As  the  latter  is  also  a condition  not  often  found  in 
commercial  service,  it  may  be  said  that,  in  general,  curves  of 
impressed  and  active  voltages  are  neither  similar  in  form  nor 
coincident  in  phase;  but  they  are  always,  perforce,  of  the 
same  frequency,  though  usually  composed  of  more  than  one 
harmonic. 


Fig.  377.  — Diagram  arranged  to  show  the  Angular  Space  Position  of  Alternator  Pole 
Cores  with  the  Angular  Time  Value  of  Voltages  or  Currents. 

The  curves  are  usually  plotted  to  rectangular  coordinates; 
magnetic  densities,  magnetic  linkages,  instantaneous  voltages, 
or  instantaneous  currents  being  plotted  as  ordinates,  and  elec- 
trical degrees  or  time  as  abscissas.  To  more  definitely  locate 
the  phases  it  is  not  unusual  to  indicate  the  position  of  the  pole 
pieces  by  laying  them  off  at  the  top  or  the  bottom  of  the  plot 
(Fig.  377),  a method  used  heretofore  in  this  text. 

To  make  a complete  study  of  the  magnetic  flux  distribution 
in  an  alternator,  the  machine  should  be  worked  at  various 
loads  under  different  conditions  of  current  lag.*  Since,  at  any 
instant,  the  effect  of  the  armature  current  on  the  magnetization 


* Art.  155. 


SYNCHRONOUS  MACHINES 


t>43 


of  the  pole  pieces  depends  upon  the  positions  of  the  armature 
conductors  as  well  as  the  strength  of  the  current,  the  effect  of  the 
magneto-motive  force  of  the  armature  is  evidently  a variable, 
and  consequently  the  distribution  of  the  magnetic  flux  will  not 
be  constant  in  a single-phase  machine.  In  other  words,  both 
the  cross  and  back  magneto-motive  force  of  the  armature  wind- 
ing for  any  load  vary  continuously  during  each  period,  and 
therefore  the  magnetic  flux  distribution  varies  with  the  position 
of  the  armature.  The  variation  due  to  this  cause  is  probably  not 
very  great  in  machines  with  multitooth  armatures,  and  an  aver- 
age distribution  may  be  assumed  as  satisfactorily  representing 
working  conditions.  In  polyphase  machines  where  the  system 
is  balanced,  the  magneto-motive  forces  of  the  windings  of 
the  several  phases  of  the  armature  combine  to  form  an  approxi- 
mately constant  reaction;  unbalancing,  however,  destroys  the 
uniformity  of  the  resultant  and  the  reactions  vary  in  value, 
depending  upon  the  amount  of  unbalancing.*  In  machines 
with  only  a few  teeth,  the  effect  of  armature  reactions  and  the 
movement  of  the  teeth  across  the  pole  faces  is  often  sufficient 
to  cause  regular 
pulsations  in  the 
field  flux  and  ex- 
tremely marked 
distortions  in  the 
voltage  curve. 

The  pulsations  are 
sometimes  suffi- 
cient to  materially 
affect  the  exciting 
current.  Figure 
378,  Curve  II, 
shows  an  experi- 
mentally deter- 

n " j,  , i Fig.  378.  — Curve  illustrating  tne  Variation  of  Field 
mined  curve  of  the  Current  caused  by  Poorly  Designed  Armature  Teeth, 
field  current  of  a Curve  I is  the  Irregular  Voltage  Wave,  and  Curve  II 
shows  the  Pulsations  in  the  Field  Current  due  to  the 
Reactions  of  the  Teeth  and  the  Armature  Current. 

nator  having  a very 

deeply  toothed  armature  core  with  one  tooth  per  coil  and  a large 
self-inductance  in  the  armature.  The  machine  was  excited  by  a 

* Art.  152. 


single-phase  alter- 


644 


ALTERNATING  CURRENTS 


small  shunt-wound  exciter.*  The  very  irregular  wave  of  vol- 
tage is  shown  by  curve  I. 

Various  methods  may  be  used  for  experimentally  determin- 
ing the  form  of  electric  voltage  or  current  curves. 

1.  The  Oscillograph.  — Much  the  most  efficient  method  of 
tracing  alternating  current  and  voltage  curves  is  by  means 
of  the  Oscillograph,  as  originally  worked  out  by  Blondel  and 
Duddell,  but  now  become  one  of  the  important  measuring  in- 
struments for  alternating  current  phenomena.  This  instrument 


Fig.  379.  — Diagrams  showing  the  Essential  Features  of  a Three-element  Oscillo- 
graph. I,  View  from  the  Side.  II,  View  from  Above. 

in  essential  parts  consists  of  a loop  of  wire  ribbon  placed  in  a 
strong  field  of  constant  magnetic  flux  with  the  plane  of  the  loop 
parallel  to  the  lines  of  force  of  the  field.  Figure  379  is  a dia- 
gram of  the  optical  system  of  an  oscillograph  having  three 
independent  loops  placed  at  V V,  and  V.  Sketch  I is  a side 
view  and  II  a view  looking  from  above.  The  curve  desired 
may  either  be  made  visible  upon  the  screen  D or  be  photographed 
on  a sensitized  film  wrapped  upon  the  drum  T.  First  consider- 
ing the  visible  record:  an  arc  light  at  F throws  a beam  of  light 


D 


* See  Trans.  Amer.  Inst.  E.  E.,  Vol.  7,  p.  374. 


SYNCHRONOUS  MACHINES 


645 


through  the  condensing  lens  E to  the  prisms  _P,  _P,  P , where  it 
is  deflected  in  three  beams  and  passes  to  mirrors  fastened  to  the 
loops  at  E,  V,  V,  each  mirror  being  fastened  at  one  edge  to  one 
strand  of  a loop  and  at  the  other  edge  to  the  other  strand  of  the 
same  loop.  The  loops  are  so  designed  that  the  two  wires 
of  the  sides  have  a very  rapid  natural  rate  of  vibration  — some- 
times as  much  as  10,000  vibrations  per  second,  though  more 
frequently  about  one  half  that  number.  The  beams  of  light 
thrown  upon  the  mirrors  V,  V,  V pass  through  a long  cylindrical 
lens  at  C1  and  fall  upon  a long  mirror  shown  at  B which  oscil- 
lates on  a horizontal  axis.  At  the  focus  of  C is  placed  a semi- 
opaque screen  D.  When  an  alternating  current,  the  curve  of 
which  is  to  be  determined  is  passed  through  a loop,  the  magnetic 
reactions  between  the  current  in  the  two  sides  of  the  loop  and  the 
constant  field  flux  tend  to  turn  the  mirror  at  V to  the  left  or 
right,  depending  upon  the  instantaneous  direction  of  the  current. 
Since  the  natural  rate  of  vibration  of  the  loop  is  much  higher 
than  that  of  the  current  or  its  harmonics,  the  curve  of  which 
is  to  be  traced,  the  mirror  changes  its  angular  position  substan- 
tially proportionately  to  the  instantaneous  values  of  the  current. 

The  beams  of  light  from  V,  V,  V move  over  the  length  of 
mirror  B.  Mirror  B is  oscillated  at  one  half  synchronous  speed 
compared  with  the  frequency  of  the  current  through  the  loop 
wires.  The  beams  of  light  are  cast  by  B upon  the  semiopaque 
screen  at  JD.  Each  is  then  subjected  to  the  resultant  of  the 
two  mirror  movements,  i.e.  a movement  by  B which  is  uniform 
and  is  proportional  to  the  angular  advance  of  the  voltage  of 
the  circuit,  and  a movement  at  right  angles  thereto  caused  by 
the  mirror  at  V,  V,  V,  which  is  at  each  instant  proportional  to 
the  strength  of  current  in  the  loop  wires.  The  beams  of  light 
thus  trace  upon  the  screen  curves  in  which  the  abscissas  caused  by 
B are  always  proportional  to  the  angular  advance  during  a cycle 
of  the  impressed  voltage  and  in  which  the  ordinates  are  propor- 
tional at  each  instant  to  the  current  strength  in  the  loop  wires. 
The  curves  are,  therefore,  of  the  same  character  as  those  obtained 
by  the  ordinary  method  of  plotting  alternating  current  waves. 

By  using  the  several  oscillograph  loops  at  once  and  casting 
the  beams  of  light  from  their  mirrors  upon  the  same  mirror  B, 
curves  of  three  currents  may  be  traced  upon  the  screen  at  the 
same  time  and  their  relative  phase  positions  and  forms  compared. 


646 


ALTERNATING  CURRENTS 


The  wave  cycles  are  repeated  over  and  over  at  the  frequency  of 
the  impressed  voltage,  and  for  ordinary  frequencies  appear  like 
fixed  curves  on  the  semiopaque  screen.  To  prevent  the  curve 
from  being  shown  double,  the  shutter  S is  driven  in  synchron- 
ism with  the  vibrating  mirror  B in  such  a way  as  to  cut  off  the 
light  from  F during  every  other  half  oscillation  of  B. 

When  the  photographic  apparatus  is  to  be  used  the  beams  of 
light  from  V,  V,  V pass  through  the  long  cylindrical  lens  C2  and 
strike  the  photographic  film  placed  upon  T,  (7a  and  B being 
moved  aside.  The  drum  T is  rotated  at  a desirable  speed  and 
the  shutter  S opens  and  closes  at  the  beginning  and  end  of  one 
of  its  revolutions.  The  movement  of  the  drum  obviously  de- 
scribes the  abscissas  of  the  curve. 


Fig.  380.  — Vibrating  Mechanism  and  Electro-magnets  of  One  Type  of  Oscillograph  — 

Cover  removed. 

Figure  380  shows  the  vibrators  and  magnets  of  a commercial 
type  of  three  element  oscillograph.  In  this  figure,  Gr  is  one 
of  the  openings  for  admitting  the  three  beams  of  light  to  the 
galvanometer  or  loop  mirrors  just  inside.  B is  one  of  the  elec- 
tromagnets. The  cells  holding  the  loops,  sometimes  called 
vibrators,  one  of  which  is  shown  at  W.  are  immersed  in  a liquid 
to  deaden  the  natural  vibrations. 


SYNCHRONOUS  MACHINES 


647 


Figure  381  shows  the  vibrator  element  at  A,  which  is  made 
of  fine  silver  or  bronze  ribbon.  The  part  between  the  bridges 
B and  B1  is  between  the  magnet  poles  and  to  that  part  is  at- 
tached the  mirror. 

To  obtain  curves  of  current,  current  transformers  may  be  re- 
quired, or  sometimes  the  loop  is  shunted  around  a non-reactive 
resistance  device  through  which  the  current  to  be  measured 
flows.  In  measuring  voltages  it  is  in  ordinary  cases  necessary 
to  use  a constant  voltage  trans- 
former to  cut  down  the  voltage 
impressed  upon  the  loop  as  well 
as  to  place  non-inductive  resist- 
ance in  series  in  the  loop  circuit. 

A forerunner  of  the  oscillo- 
graph is  due  to  Gerard  and  may 
sometimes  be  used  for  tracing 
the  voltage  curves  of  an  alter- 
nator when  no  current  flows  in 
the  armature.  The  machine  to 
be  tested  is  rotated  at  a very  slow 
speed,  the  field  being  excited  in 
the  usual  manner.  The  terminals 
of  the  machine  to  be  examined 
are  connected  to  a shunted  d’Ar- 
sonval  galvanometer.  The  nat- 
ural rate  of  oscillation  of  the 
galvanometer  bobbin  is  made 
quite  rapid,  as  compared  with 
the  period  of  the  voltage  supplied 
by  the  alternator  at  its  slow 
speed.  Then  the  deflection  of  the  needle  at  each  instant  will 
be  proportional  to  the  instantaneous  voltage.  By  moving  a 
sheet  of  sensitized  paper  before  the  galvanometer  mirror  which 
throws  upon  it  a beam  of  light,  the  paper  being  moved  at  right 
angles  to  the  mirror  movement,  the  curve  of  voltage  may  be 
permanently  recorded.* 

Figure  382  is  a current  curve  traced  by  an  oscillograph  of  a 
current  containing  a marked  peak  caused  by  highly  saturated 
iron  cores  in  coils  in  series  with  the  circuit.  Figure  383  is  a 


Fig.  381.  — An  Oscillograph  Vibrator. 


* See  Gerard’s  Le$ ons  sur  l' Electricite,  Vol.  1,  p.  565,  3d  ed. 


648 


ALTERNATING  CURRENTS 


Fig.  382.  — Peaked  Current  Curve  traced  by  an  Os- 
cillograph. 


voltage  curve  of  an 
alternator  which  is 
almost  sinusoidal  in 
form.  In  both  cases 
the  waves  were  taken 
by  means  of  the  pho- 
tographic attachment 
to  the  instrument. 

2.  3Iethods  using 
Contact  Makers.  — 
A large  number  of 
methods  require  the 
use  of  a revolving 
contact  maker  of 
some  kind,  and  they 
therefore  have  much 
in  common.  These 
were  much  used  in 
the  past  and  are  use- 
ful for  special  pur- 


poses or  where  an  os- 
cillograph is  not  avail- 
able. The  principal 
differences  in  the 
methods  relate  to  the 
types  of  instruments 
used  to  give  the  indi- 
cations, and  the  con- 
venience with  which 
the  manipulations  may 
be  made.  Whether 
current  or  voltage 
curves  are  to  be  ob- 
tained, only  instanta- 
neous voltage  measure- 
ments are  made.  For 
the  former,  the  instan- 
taneous voltages  ai’e 
taken  at  the  terminals 


Fig.  383.  — Sinusoidal  Alternator  Voltage  Curve  traced 
by  au  Oscillograph. 


SYNCHRONOUS  MACHINES 


649 


of  non-reactive  resistance,  and  the  instantaneous  currents  are 
readily  deduced. 

In  one  of  the  earlier  methods,  advanced  and  used  by  Joubert, 
the  terminals  of  the  alternator  armature,  or  of  one  bobbin  of 
the  armature,  are  recurrently  connected  to  a condenser  in  the 
following  manner  : One  armature  terminal  is  connected  perma- 
nently to  one  terminal  of  the  condenser;  the  other  armature 
terminal  is  connected  to  a rotating  point  which  may  be  put  into 
brief  connection  with  the  free  terminal  of  the  condenser  when 
the  armature  is  at  any  desired  point  in  its  rotation.  At  the 
instant  this  contact  is  made,  the  condenser  receives  a charge 
which  is  proportional  to  the  instantaneous  voltage  between  the 
terminals.  The  charge  may  be  measured  by  discharging  the  con- 
denser through  a ballistic  galvanometer  and  the  voltage  computed 
from  the  amount  of  charge  and  the  capacity  of  the  condenser. 
By  successively  setting  the  contact  maker  so  that  the  instant 
of  contact  corresponds  with  various  points  in  the  revolution  of 
the  armature,  the  correspond- 
ing instantaneous  voltages  may 
be  thus  measured  and  the  curve 
of  voltage  may  be  plotted  (Fig. 

384).  The  contact  maker  used 
by  Joubert  was  an  insulated 
pin  set  in  the  armature  shaft, 
against  which  a brush  could  be  Fig.  384.  — Approximate  Curve  of  Voltage 
made  to  bear  at  any  point  in  the  obtained  by  the  Use  of  a Contact  Maker. 

revolution.  A quadrant  electrometer  may  be  used  in  place  of 
the  condenser  and  ballistic  galvanometer ; in  which  case  it  is 
desirable  to  introduce  a condenser  permanently  in  parallel  with 
the  electrometer,  to  neutralize  the  effect  of  leakage  in  the  test 
circuit. 

Joubert’s  investigations  made  in  1880  resulted  in  the  first 
determination  of  the  curve  of  voltage  of  an  alternator.  The 
investigations  of  Duncan,  Hutchinson,  and  Wilkes  probably 
produced  the  earliest  series  of  experimental  curves  showing  the 
relations  between  the  waves  of  voltage  and  current  in  circuits 
of  different  kinds.  The  investigations  of  Searing  and  Hoffman 
were  probably  the  first  made  upon  an  alternator  with  iron  in  the 
armature  core.  Their  results  showed  the  curve  of  voltage  de- 
veloped in  a smooth-core  drum  armature  with  wide  coils  to 


650 


ALTERNATING  CURRENTS 


approach  a sinusoid.  Ryan  and  Merritt,  in  1889,  did  much 
valuable  work  in  studying  the  current  and  voltage  relations  in 
a transformer.  Various  methods  were  used  and  desirable  data 
obtained  by  Blondel,  Searing,  Hoffman,  Mershon,  Duncan,  Ayr- 
ton, Pupin,  and  others.  The  last  named  used  a method  for 
determining  harmonics  by  means  of  resonance,  while  Professor 
Ayrton  proposed  obtaining  the  harmonics  by  means  of  the  vibra- 
tions of  a stretched  wire. 

The  earliest  and  simplest  contact  maker  was,  as  already 
pointed  out,  simply  an  insulated  pin  set  in  the  shaft  of  the 
alternator  furnishing  the  current  for  the  test.  With  this  was 
a brush  so  arranged  as  to  make  contact  with  the  pin  at  any 
desired  point  in  the  revolution.  This  arrangement  is  often 
inconvenient  of  application,  and  is  likely  to  give  rather  irregular 
results.  The  contact  is  likely  to  be  variable  in  resistance  and, 
as  the  brush  wears,  the  duration  of  contact  varies.  Each  of 
these  variables  introduces  errors  of  greater  or  less  magnitude, 
depending  upon  the  conditions  of  the  test.  Various  refine- 
ments of  construction  have  been  introduced  by  experimenters 
in  order  that  the  defects  of  the  contact  makers  may  be  eliminated. 
If  a contact  of  absolute  uniformity  were  assured,  special  instru- 
ments would  not  be  necessary  for  taking  the  indications  in 
determining  voltage  and  current  curves,  because  the  indications 
of  a sensitive  electrodynamometer  might  then  be  directly  used. 

The  contact  maker  was  originally  arranged  for  a single  con- 
tact, but  it  is  frequently  desirable  to  make  simultaneous  observa- 
tions of  several  curves.  This  may  be  readily  accomplished  by 
using  a contact  maker  with  the  appropriate  number  of  contact 
disks  on  the  same  spindle.  Then  a satisfactory  instrument, 
such  as  an  electrostatic  voltmeter,  may  be  used  in  each  circuit. 


Fig.  385.  — • Contact  Maker  with  a Flexible 
Shaft. 


Sometimes  it  is  not  conven- 
ient to  have  the  contact 
maker  attached  to  the  dy- 
namo shaft,  in  which  case  it 
may  be  attached  to  a short 
length  of  flexible  shaft  (Fig. 
385),  which  may  in  turn  be 
attached  to  the  djnamo 
shaft,  but  the  flexible  shaft 
must  possess  uniform  tor- 


SYNCHRONOUS  MACHINES 


651 


sional  rigidity  or  the  rate  of  rotation  of  the  contact  maker  may 
be  unsteady,  or  the  instrument  may  “ hunt,”  and  thereby  in- 
troduce error  by  causing  variations  of  the  contact  duration. 
When  connection  to  the  alternator  cannot  be  conveniently 
made,  the  contact  maker  may  be  driven  by  a synchronous  motor 
as  has  been  done  by  Blondel,  Siemens  and  Halske,  and  Fleming. 
In  this  case,  “ hunting  ” by  the  synchronous  motors  is  likely  to 
cause  errors. 

The  contact  maker  shown  in  Fig-  385  indicates  the  attach- 
ment of  the  contact  brush  to  a scale,  whereby  it  can  be  moved 
to  any  desired  angular  position  and  there  make  contact  once  in 
a revolution  with  the  contact  button  on  the  revolving  drum. 

The  plan  of  charging  and  discharging  a condenser  to  measure 
the  instantaneous  voltage  is  not  convenient,  and  various  other 
expedients  have  been  proposed  and  more  or  less  used.  A gal- 
vanometer with  a sufficiently  great  natural  rate  of  vibration  will 
be  steadily  deflected  by  the  succession  of  impulses  which  it  re- 
ceives when  connected  in  circuit  with  a contact  maker.  This 
deflection  may  be  balanced  by  a steady  voltage  which  is  intro- 
duced in  the  circuit  in  series  with  the  galvanometer  and  contact 
maker  as  indicated  in  Fig.  386.  A telephone  put  in  the  place 


-Jo- 

CONTACT  MAKER 


REVERSING  KEY 


© 


GALVANOMETER 


of  the  galvanom- 
eter gives  oral  - 
instead  of  ocular 
notice  of  a bal- 
ance. It  is  de- 
sirable to  place  a 
condenser  in  par- 

allQl  with  the  gal  ^IG'  — Potentiometer  Arrangement  for  Contact  Maker. 

vanometer  or  telephone  when  this  arrangement  is  used.  In 
Fig.  386,  cd  represents  a graduated  rheostat  with  a moving 
contact  post  e,  V is  a voltmeter,  and  B a battery  or  direct  cur- 
rent dynamo,  to  give  a steady  voltage  which  is  greater  than 
the  maximum  instantaneous  voltage  to  be  observed.  The  con- 
tact post  c is  moved  along  the  rheostat  until  a balance  is 
produced,  and  the  voltmeter  reading  is  then  equal  to  the  instan- 
taneous voltage  corresponding  to  the  particular  setting  of  the 
contact  maker.  The  reversing  key  is  introduced  in  the  circuit 
to  make  it  equally  convenient  to  explore  the  positive  and  nega- 
tive loops  of  the  alternating  voltage. 


652 


ALTERNATING  CURRENTS 


Any  method  enabling  the  use  of  a reflecting  instrument  in 
circuit  with  the  contact  maker  may  be  made  continuously  self- 
recording  by  a proper  disposition  of  the  apparatus.  In  this 
case,  a beam  of  light  is  thrown  upon  the  mirror,  and  its  devia- 
tion is  recorded  by  means  of  a moving  photographic  film.  In 
order  that  the  complete  curve  may  be  thus  recorded,  the  contact 
points  must  be  caused  to  rotate  continuously  around  the  spindle 
of  the  contact  maker.  Since  the  needle  of  the  galvanometer 
or  electrometer  which  is  used  with  the  contact  maker  in  this 
case  must  rigidly  follow  the  intensity  of  the  current  impulses, 
the  instrument  must  have  little  inertia  and  be  truly  deadbeat. 
The  vibrations  of  a telephone  diaphragm  with  a mirror  mounted 
on  it  have  been  used  to  replace  the  deviations  of  a galvanometer 
or  electrometer  needle. 

158.  Areas  of  Successive  Curves  of  Alternating  Currents  and 
Voltages.  — In  general,  observations  which  cover  one  complete 
period  entirely  define  the  curves  of  commercial  alternating  cur- 
rents and  voltages.  Since  there  is  no  continuous  transference  of 
electricity  in  one  direction,  the  areas  of  successive  loops  of  the 
curves  should  be  equal.  In  the  voltage  curves  produced  by 


an 


alternator,  for  instance,  e — and  = Cedt , where  is 

dt  J 


the  total  number  of  lines  of  flux  passing  into  the  armature  core 
and  T is  the  time  occupied  by  a cycle.  If  and  Tare  constant, 
as  would  be  the  case  for  a symmetrical  alternator  with  fixed 
field  magnetism  and  a rigid  armature  shaft,  which  is  driven  at 
a uniform  speed,  the  areas  of  the  successive  loops  of  the  voltage 
curve  must  be  equal.  On  account  of  various  irregularities  in 
the  construction  and  working  of  alternators,  experimentally 
determined  curves  are  not  always  uniform.  In  fairly  large 
commercial  machines  the  differences  are  usually  not  greater 
than  might  be  caused  by  the  errors  of  observation  due  to  the 
experimental  determination ; and  appreciable  differences  in  the 
areas  of  successive  loops  of  the  curves  produced  by  mechani- 
cally rigid  machines,  driven  at  a uniform  angular  velocity,  are 
not  to  be  expected,  except  possibly  when  the  machines  have 
armatures  with  their  halves  connected  in  parallel,  and  then 
only  when  the  magnetic  circuits  lack  symmetry  to  a consider- 
able degree.  In  the  case  of  certain  small  eight-pole  alternators, 
Dr.  Bedell  found  differences  in  the  areas  of  the  consecutive 


SYNCHRONOUS  MACHINES 


653 


loops  which  are  not  explainable  upon  the  ground  of  errors  of 
observation  or  of  variable  speed.*  The  curves  given  by  two 
of  these  machines  in  one  complete  revolution  (four  complete 
periods)  are  shown  in  Fig.  387.  The  individual  areas  of  the 


loops  are  marked  upon  the  figure.  While  these  differ  as  much 
as  25  per  cent  amongst  themselves,  the  sums  of  the  positive 
and  negative  areas  differ  by  no  more  than  might  be  caused  by 
* Physical  Review , Vol.  1,  p.  218, 


654 


ALTERNATING  CURRENTS 


experimental  errors,  while  the  angular  speed  of  the  prime 
mover  or  of  the  exciter  may  cause  similar  deviations  in  the 
curves.  This  apparently  shows  that  irregularities  in  the  mag- 
netic circuits  and  in  the  armature  windings  may  in  some  cases 
cause  differences  in  the  successive  loops  of  the  curve  developed 
in  360  electrical  degrees,  but  the  algebraic  summation  of  the 
areas  due  to  each  revolution  is  zero.  The  latter  must  be  true, 
or  there  would  be  a continuous  flow  of  electricity  in  one  direc- 
tion. The  fact  that  the  machines  tested  by  Dr.  Bedell  had 
notable  structural  weaknesses  leads  to  the  probability  that  the 
springing  of  the  shaft  or  other  parts  of  the  machine  may  have 
caused  the  unusual  result  which  he  found.  Indeed,  it  is  not 
uncommon  where  an  alternator  is  badly  aligned  or  the  shaft  is 
slightly  bent,  in  either  the  alternator  or  its  exciter,  to  have  such 
irregularities  set  up  in  the  voltage  wave.  Irregular  angular 
speed  of  prime  movers  or  of  the  exciters  may  cause  similar 
variations  of  the  areas.  These  may  become  so  pronounced  in 
certain  voltage  loops  occurring  during  the  alternator  revolu- 
tions, as  to  be  manifested  to  the  eye  by  flickering  of  incandes- 
cent lamps  attached  to  the  alternator  circuit.  Variation 
between  the  areas  of  the  successive  loops  of  current  curves  may 
be  introduced  by  unsteady  resistance  or  reactance,  even  when 
the  voltage  curve  is  perfectly  uniform. 

159.  Regulation  of  Alternators  for  Constant  Voltage.  — A sep- 
arately excited  alternator  has,  as  already  intimated,  no  inherent 
tendency  towards  regulation.  The  regulation  is  usuall}'  effected 
by  means  of  a rheostat  in  the  field  circuit  of  the  shunt  or  com- 
pound-wound exciter,  a rheostat  in  series  with  the  alternator 
field  winding,  or  both.  The  adjustment  of  these  rheostats 
may  be  performed  by  hand  or  through  devices  actuated  by  a 
relay  placed  in  shunt  to  the  main  circuit.  In  Great  Britain 
and  Europe  automatic  devices  have  been  used  in  large  plants 
for  many  years,  and  of  recent  years  they  have  come  into  very 
common  use  in  this  country. 

A type  commonly  used  in  this  country  for  controlling  alter- 
nator voltage  is  arranged  to  make  brief  successive  periods  of 
short  circuit  around  a part  or  all  of  the  field  rheostat  of  the  ex- 
citer, the  periods  of  short  circuit  being  of  greater  or  less  duration 
depending  upon  whether  the  alternator  voltage  should  be  in- 
creased or  decreased.  The  short-circuiting  periods  repeat  them- 


SYNCHRONOUS  MACHINES 


655 


selves  in  rapid  succession,  varying  the  field  current  and  giving  it  a 
wavy  form  having  a magneto-motive  force  of  the  value  needed 
to  set  up  the  required  exciter  voltage.  The  exciter  field  rheostat 
is  usually  set  so  that  when  the  short  circuit  around  the  rheostat 
has  been  opened  the  alternator  voltage  will  fall  in  from  six  to 
eight  seconds  to  about  80  per  cent  of  normal  full  load  value 
when  compound  or  interpole  exciters  are  used,  and  to  the  no- 
load  alternator  voltage  when  shunt  exciters  are  used.  The 
delay  with  which  the  voltage  falls  to  its  lower  value  when  the 
rheostat  comes  into  the  exciter  field  circuit  by  the  removal  of 
the  short  circuit  is  caused  by  the  eddy  currents  and  the  self- 
inductance causing  an  apparent  electro-magnetic  inertia  in  the 
exciter  and  alternator  field  magnets. 

At  other  than  no  load  the  short-circuiting  contact  of  a regu- 
lator of  this  type  seems  always  rapidly  moving.  The  smaller 
the  load  and  the  greater  the  resultant  tendency  for  the  line 
voltage  to  rise,  the  shorter  must  be  the  periods  of  the  short 
circuit  of  the  exciter  field  rheostat.  Figure  388  shows  a dia- 

D CURRENT 


Fig.  388.  — Voltage  Regulator  which  short-circuits  a Siugle  Section  of  the  Exciter 
Field  Rheostat  and  has  One  Set  of  Relay  Contact  Parts. 

gram  of  one  of  the  commercial  types  of  such  a regulator.  The 
regulator  has  a direct  current  control  magnet  A,  an  alternating 
current  control  magnet  B,  and  a relay  C.  The  winding  of  the 
magnet  A is  connected  across  the  exciter  circuit,  and  its  movable 
core  is  attached  to  one  end  of  a pivoted  lever  which  carries  at 
its  opposite  end  a flexible  contact  part  D which  is  pulled  down- 
ward by  springs.  The  magnet  B has  two  windings,  one  being 
connected  to  a potential  transformer  and  the  other  being  a 


65b 


ALTERNATING  CURRENTS 


compensating  winding  connected  to  a current  transformer  and 
giving  an  opposing  magnetizing  effect  proportional  to  the  cur- 
rent flowing  in  the  alternator  circuit.  The  movable  core  of 
magnet  B is  attached  to  one  end  of  a pivoted  lever  which  is 
counterweighted  at  its  other  end  and  carries  a contact  part 
which  cooperates  with  the  contact  part  D already  referred  to, 
securing  what  may  be  called  a floating  contact  arrangement. 
The  relay  C consists  of  a U-shaped  magnet  core  carrying  a 
differential  winding  and  provided  with  a pivoted  armature 
which  controls  a circuit  contact  arranged  to  close  and  open  the 
branch  which  short-circuits  the  exciter  field  rheostat.  One 
half  of  the  differential  winding  is  permanently  connected  across 
the  exciter  circuit.  The  other  half  of  the  differential  winding 
is  connected  to  the  floating  contact  points,  and  is  also  con- 
nected across  the  exciter  circuit  and  neutralizes  the  magnet- 
izing effect  of  the  first  half  when  the  floating  contact  is 
closed.  A condenser  K is  connected  between  the  contact  parts 
at  the  relay  armature,  to  prevent  serious  sparking  when  the 
contact  opens. 

Now,  assuming  the  floating  contact  to  be  open,  the  relay 
armature  is  drawn  down  and  the  field  rheostat  is  introduced 
in  the  exciter  field  circuit ; this  lowers  the  voltage  of  the  ex- 
citer and  thereby  lowers  the  voltage  of  the  alternating  current 
generator  to  which  the  exciter  is  connected.  This  weakens 
both  of  the  control  magnets,  and  the  floating  contact  closes, 
whereupon  the  second  half  of  the  differential  winding  of  the 
relay  is  energized  and  neutralizes  the  relay  excitation,  and  the 
relay  armature  is  released.  This  results  in  closing  the  relay 
contact  and  short-circuiting  the  field  rheostat.  That  increases 
the  excitation  of  the  exciter  and  therefore  of  the  alternator,  so 
that  both  exciter  and  main  voltages  increase.  Both  control 
magnets  are  strengthened,  the  core  of  A is  attracted  downward 
and  the  core  of  B is  attracted  upward  so  that  the  floating  con- 
tact opens,  interrupting  the  circuit  through  one  half  of  the 
differential  winding  on  (7,  the  relay  armature  is  attracted,  and 
the  exciter  field  rheostat  is  again  short-circuited.  As  the  vol- 
tages rise  in  consequence  of  the  last  condition,  the  cycle  of 
operation  begins  again,  and  it  is  repeated  over  and  over  again 
at  a fairly  high  rate  of  vibration  which  maintains  the  alternator 
voltage  with  reasonable  precision  at  the  value  desired  for  all 


SYNCHRONOUS  MACHINES 


657 


loads.  The  actual  variation  of  voltage  required  to  produce  the 
cycle  of  operations  is  very  slight. 

It  is  often  desirable  to  have  the  regulator  control  the  alter- 
nator so  as  to  keep  the  voltage  constant  at  the  point  of  con- 
sumption rather  than  at  the  generating  plant,  and  this  may 
be  reasonably  accomplished  by  the  use  of  the  combination  of 
current  transformer  and  potential  transformer  indicated  in  Fig. 
388.  This  obviates  the  use  of  “ pressure  wires  ” (that  is,  wires 
which  run  from  the  center  of  consumption  to  voltmeters  at  the 
generating  station  for  the  purpose  of  indicating  the  voltage  at 
the  consumption  center),  which  were  much  used  in  the  early 
days  of  electric  lighting. 

Where  several  exciters  are  used  in  parallel  on  the  exciter  bus 
bars  they  may  be  connected  as  shown  in  Fig.  389.  In  this  case, 


Fig.  389.  — Several  Exciters  connected  to  a Voltage  Regulator  of  the  Type  in  which 
Part  of  the  Field  Rheostats  are  short-circuited. 


the  alternator  field  windings  are  connected  in  parallel,  and  care 
must  be  taken  that  the  regulation  affects  the  voltages  of  all  the 
exciters  alike,  or  the  exciterloads  will  not  divide  equally.  This 
requires  care  in  the  adjustment  of  the  rheostats,  and  of  the  exciter 
field  compounding  where  the  exciters  are  compound  wound. 
When  the  exciters  are  of  large  capacity,  the  relay  magnet  controls 
several  sets  of  contact  parts  so  that  each  exciter  rheostat  can  be 
operated  by  an  independent  set,  or  two  or  more  rheostats  con- 
2 u 


658 


ALTERNATING  CURRENTS 


trolled  by  separate  contacts  can  be  used  for  the  field  winding  of 
a single  exciter.  By  such  an  arrangement  the  values  of  the  cur- 
rents and  voltages  can  be  reduced  suitably  to  be 
handled  by  the  contacts  without  injury. 

The  device  used  for  obtaining  at  the  termi- 
nals of  the  regulator  a voltage  which  is  propor- 
tional to  1Z , where  E is  the  generator 

terminal  voltage  and  IZ  is  the  line  drop,  is 
similar  in  operation  to  the  arrangements  used 
for  obtaining  compensated  voltmeter  readings. 
It  is  often  desirable  to  show  at  the  switch-board 
the  voltage  at  the  point  of  consumption.  The 
station  voltmeter  is  then  connected  in  series 
with  the  secondary  windings  of  a voltage  trans- 
former and  a current  transformer  which  act 
in  opposition  (Fig.  390).  The  transformers 
being  properly  adjusted,  the  voltmeter  shows 
at  all  times  the  voltage  at  the  center  of  con- 
sumption. In  such  an  arrangement  several 
terminals  may  be  brought  out  from  the  sec- 
ondary winding  of  the  current  transformer  to  a 
switch  like  that  shown  in  Fig.  391,  and  the  ad- 
justment of  the  apparatus  to  compensate  for  any 
given  line  drop  may  then  be  made  by  changing 
the  secondary  connections  by  moving  the  switch 
lever  and  so  changing  the  number  of  effective  secondary  turns 
in  the  voltmeter  circuit.  The  secondary  winding  of  the  cur- 
rent transformer  may  be  shunted  by  resistance 
as  at  R2  in  Fig.  392,  and  the  adjustment  of  this 
resistance  serves  a corresponding  purpose.  To 
get  exact  compensation  in  the  voltmeter  circuit, 

E2  should  be  replaced  by  resistance  and  react- 
ance in  the  same  proportions  as  the  resistance 
and  reactance  of  the  line  for  the  drop  over 
which  compensation  is  to  be  made.  If  the  ratios 
of  transformation  of  the  potential  and  current 
transformer  T and  T'  are  alike,  the  ratio  of  the 
line  impedance  to  the  impedance  substituted  in 
the  position  of  E2  should  be  equal  to  the  ratio  of  transformation, 
and  the  voltage  at  the  far  end  of  the  line  will  then  bear  the  same 


Fig.  390.  — Diagram 
of  Connections  for 
a Compensated 
Voltmeter. 


Fig.  391.  — Hand- 
regulating Switch 
for  Controlling 
Number  of  Turns 
in  Transformer 
Secondary. 


SYNCHRONOUS  MACHINES 


659 


ratio  to  the  voltage  measured  by  the  voltmeter.  If  the  ratio  of 
transformation  of  the  current  transformer  differs  from  the  ratio 
of  the  potential  transformer,  then  the  shunting  impedance 
should  be  changed  in  a corresponding  degree.  The  secondary 
winding  of  the  current  transformer  does  not  necessarily  have 
to  be  connected  into  the  circuit  of  the  regular  voltmeter  coil, 
but  can  be  connected  to  an  auxiliary  coil  which  is  wound  along- 
side of  or  over  the  main  coil. 

When  a regulator  without  a compensating  coil  is  used,  the 
voltage  acting  in  the  circuit  of  the  automatic  regulator  must  be 
caused  to  remain  constant  as  the  alternator  current  increases, 
while  at  the  same  time  the  alternator  voltage  increases  by  a 
sufficient  amount  to  compensate  for  the  fall  of  voltage  in  the 
feeders,  in  case  the  regulator  is  intended  to  maintain  constant 
voltage  at  the  center  of  consumption.  In  other  words,  E — IZ 
must  be  kept  constant,  E being  the  alternator  voltage,  I the 
current,  and  Z the  impedance  of  the  feeders.  This  may  he 
effected  approximately  as  illustrated  in  Fig.  388,  or  as  follows 
(Fig.  392).  The  regulator  is  connected  to  the  secondary  cir- 
cuit of  potential  transformer  T,  R 

which  is  connected  in  parallel 
across  the  feeders.  The  voltage 
of  the  secondary  of  this  trans- 
former is  proportional  to  the  al- 
ternator voltage  E.  A current 
transformer  T'  is  connected  with 
its  primary  winding  in  series  with 
the  feeders.  The  voltage  devel- 
oped in  the  secondary  winding  of 
this  transformer  can  be  adjusted 
so  as  to  be  practically  proportional 
to  IZ  for  all  values  of  the  current. 

The  secondary  winding  of  this 
transformer  is  connected  in  series 
with  the  secondary  circuit  of  the 
first  transformer,  and  in  such  a way  that  their  voltages  are  in 
opposition.  Hence  a voltmeter  V connected  across  the  ter- 
minals of  the  two  secondaries  indicates  a voltage  which  is  pro- 
portional to  the  voltage  at  the  terminals  of  the  feeder,  or 
E — IZ.  If  the  automatic  regulator  is  also  connected  across 


1 

pv  ' 

“CT 

R3  Jh 
— fjjjj— •'Tnnnr— 

SHUNT 

Wwv 



l/VW  SER 

T,( 

WV\A 

[p  V 

q 

Fig.  392.  — Connections  of  Voltage  and 
Current  Transformer  to  Alternator 
Regulator  to  maintain  Constant  Vol- 
tage at  the  Center  of  Distribution. 


660 


ALTERNATING  CURRENTS 


the  terminals  of  the  two  secondaries,  it  adjusts  the  excitation 
of  the  alternator  so  that  E — IZ  is  kept  fairly  constant  regard- 
less of  the  value  of  I.  In  the  figure,  S is  the  solenoid  of  the 
regulator,  Rv  is  the  resistance  automatically  controlled  by  the 
solenoid  to  vary  the  excitation  of  the  alternator,  and  Rv  Rv  R3 
are  impedances  in  circuit  with  the  regulator  which  are  used 
for  adjusting  it  to  give  proper  indications  for  various  values 
of  I and  Z. 

Regulation  of  generator  voltage  by  means  of  compound  wind- 
ing is  difficult,  as  the  number  of  series  turns  that  give  regula- 
tion on  a non-reactive  load  evidently  may  fail  for  a reactive  load. 
It  is  evident,  therefore,  that  a compositely  excited  alternator 
will  only  regulate  on  loads  having  the  power  factor  for  which 
regulation  of  the  machine  was  adjusted.  The  ratio  of  series 
ampere-turns  per  field-magnet  pole  to  armature  ampere-turns 
per  coil  which  is  required  to  give  regulation  for  one  alternator 
of  a fixed  type  will  doubtless  give  equally  satisfactory  results 
on  machines  of  different  capacities  but  of  the  same  type'  but 
the  marked  differences  in  the  magnitude  of  the  effects  of  self-in- 
duction and  armature  reactions  in  alternators  of  different  types 
make  it  impossible  to  fix  any  ratio  that  would  even  approxi- 
mately cover  all  types  of  machines.  The  regulation  of  ordinary 
alternators  which  are  self-excited,  or  otherwise,  can  only  be  sat- 
isfactorily effected  by  means  of  variable  resistance  regulators 
placed  in  the  exciting  circuit,  or  by  varying  the  exciter  voltage. 
This  statement  includes  the  self-excited  alternators  now  on  the 
market  in  which  the  rectified  current  for  the  fields  is  obtained 
from  a special  independent  winding  placed  in  the  slots  with  the 
ordinary  windings  of  the  armature.  The  regulation  might  be 
effected  by  moving  the  brushes  on  the  rectifying  commutator, 
if  the  machine  was  self-excited,  but  only  at  the  expense  of  pro- 
hibitive sparking  and  wear. 

If  it  is  desired  to  have  an  alternator  give  constant  current, 
the  regulation  maybe  made  inherent  by  designing  the  armature 
reactions  and  self-induction  to  be  so  great  that  the  current 
cannot  rise  much  above  its  normal  value.  The  armature  should 
be  wound  to  generate  a voltage  upon  open  circuit  much  greater 
than  that  required  for  full  load,  and  hence  the  current  remains 
near  its  full  normal  value  up  to,  and  even  considerably  beyond, 
full  load.  Such  machines  can  be  worked  on  short-circuit  with- 


SYNCHRONOUS  MACHINES 


G61 


3,000 


2,000 


out  injury;  but  if  the  circuit  is  opened,  they  are  liable  to  injury 
on  account  of  the  excessive  open  circuit  voltage  breaking  down 
the  insulation.  These  machines  are  really 
worked  on  a part  of  the  characteristic  which 
is  caused  to  be  almost  vertical  on  account 
of  the  large  self-inductance  and  reactions 
of  the  armature  ; that  is,  the  number  of 
armature  conductors  is  many  times  greater 
than  comports  with  obtaining  the  maximum 
output  per  pound  of  copper  and  iron  in  the 
machine.  Constant-current  alternators  have 
sometimes  been  used  in  the  past  for  arc 
lighting  and  are  referred  to  here  to  show  the 
effect  of  their  extreme  design  on  regulation. 

The  external  characteristic  of  such  an  alter- 
nator is  given  in  Fig.  393.  This  shows  how 
the  terminal  voltage  increases  as  the  current 
decreases  relatively  little,  while  the  external 
circuit  is  changed  from  short  circuit  to  an 
equivalent  resistance  of  between  two  and 
three  hundred  ohms,  the  exciter  voltage  im- 
pressed on  the  field  windings  remaining 
constant.  The  curves  are  given  for  three 
different  excitations. 

160.  Feeder  Regulators. — Feeder  regula- 
tors for  use  in  plants  where  several  circuits 
are  fed  from  one  alternator  or  set  of  bus 
bars  are  in  quite  common  use,  as  already 
intimated.  The  regulator  may  be  a special 
transformer  (Fig.  394),  with  the  secondary 
windings  CD  in  series  with  one  feed  wire, 
and  the  primary  winding  AB  connected 
across  the  main  bus  bars.  The  voltage  in- 
duced in  CD  may  be  made  to  either  aid  or  oppose  the  alternator 
voltage  by  means  of  a reversing  switch,  X.  The  strength  of  the 
voltage  introduced  into  the  feeder  circuit  by  CD  may  be  varied  by 
changing  the  number  of  turns  of  CD  in  the  circuit,  by  means  of 
movable  contact  parts.  For  convenience,  such  contact  parts  are 
usually  arranged  on  the  arc  of  a circle  or  upon  a drum.  The  con- 
tact slider  at  F may  be  divided  into  two  parts  and  reactance 


1000 


500- 


AMPERES 

Fig.  393.  — External 
Characteristics  of  an 
Alternator  designed 
to  Produce  Constant 
Current. 


(362 


ALTERNATING  CURRENTS 


inserted  between  the  parts 
to  prevent  excessive  current 
from  flowing  when  a trans- 
former coil  is  short-circuited. 
Other  arrangements  are  used 
for  this  purpose  and  for  re- 
ducing sparking. 

Devices  in  which  the  regu- 
lation is  effected  by  varying 
the  position  of  the  primary 
and  secondary  coils  with  re- 
spect to  each  other,  or  of  the 
core  with  respect  to  both,  are 
also  used.  Thus,  a regulator 
with  armature  and  field  wind- 
ings arranged  like  those  of 
an  induction  motor*  (Fig. 
395),  with  independent  coil- 
wound  secondaries  of  the  re- 
Fig.  394.— Diagram  of  a Feeder  Regulator,  quired  number  of  phases,  may 
consisting  of  Transformer  with  Secondary  . , „ . . r i 

Winding  with  Variable  Turns  connected  Used  foi  polt  phase  feedei 
in  Series  with  the  Feeder  Circuit,  and  regulation.  Each  Set  of  Sec- 
Primary  Winding  in  Parallel  with  the  Qnd  coils  of  a pllase  is 
Feeder  Circuit.  J L 

connected  in  series  with  a 
line  conductor  of  the  feeder  to  be  controlled.  The  primary  wind- 
ings are  connected  in  mul-  

tiple  arc  on  the  main  poly- 
phase circuit.  The  pro- 
gressing or  rotating  mag- 
netic field  induced  by  the 
primary  windings,  caused 
by  the  magnetic  fluxes  of 
the  several  phases  rising 
successively  across  the  air- 
space and  at  angles  ad- 
vancing around  the  polar 
circle,  sets  up  a voltage  in 

the  secondary  windings.  Fig.  395. -Diagrammatic  Representation  of  a 
This  voltage  is  constant  Polyphase  Induction  Regulator. 

* Chap.  XII. 


GENERATOR 


BUS  BARS 


/,  REGULATOR  \ 

PRIMARY 

i 

U MOVEABLE  j 

SYNCHRONOUS  MACHINES 


663 


in  strength,  but  has  a phase  relation  with  the  feeder  voltage 
dependent  upon  the  angular  position  of  the  field  magnet  with 
respect  to  the  armature.  Thus  the  voltage  may  directly  add  to 
the  feeder  voltage,  be  subtracted  from  it,  or  combine  in  an 
intermediate 
angular  position, 
as  shown  in  Fig. 

396,  where  A 0 is 
the  generator  vol- 
tage of  one  phase 
and  AS,  AB\  Fig.  396. — Voltage  Diagram  of  a Polyphase  Induction 
AB",AB"' are  the  Regulator. 


feeder  voltages  beyond  the  regulator  for  one  phase  for  various 
positions  of  the  regulator  field  magnet  with  respect  to  its  arma- 
ture. One  core  carrying  either  the  primary  or  the  secondary 
windings  in  such  a regulator  is  mounted  in  a fixed  position, 
and  the  other  is  mounted  in  such  a manner  that  it  may  be  rotated 
such  part  of  a revolution  as  is  needed  for  sufficiently  influenc- 
ing the  feeder  voltage.  The  core  carrying  the  primary  wind- 
ings is  usually  made  the  movable  core. 

A single-phase  regulator  may  be  made  in  the  same  way, 
using  only  one  set  of  windings  respectively  on  the  primary  and 
secondary  cores.  In  this  case  zero  voltage  will  be  produced 
in  the  secondary  winding  when  its  coils  are  so  moved  as  to  be 
parallel  to  the  magnetic  flux  from  the  primary  winding,  and 
will  increase  to  maximum  value  when  the  coils  are  rotated  to 
the  right  or  left  from  this  point  until  they  are  at  right  angles 
to  the  flux.  Compensating  short-circuited  coils  such  as  are  used 
in  series  motors  * must  evidently  be  used  for  a single-phase 
regulator  of  this  kind  to  prevent  an  excessive  reactance  in  the 
secondary  circuit  when  its  windings  are  not  at  right  angles  to 
the  flux  from  the  primary  winding.  These  compensating  coils 
are  wound  on  the  core  with  the  primary  winding  but  magneti- 
cally at  right  angles  thereto.  The  transformer  characteristics 
of  these  induction  regulators  makes  their  operation  subject  to 
the  condition  that  the  vector  difference  between  the  primary 
ampere-turns  and  the  secondary  ampere-turns  plus  the  ampere- 
turns  of  any  compensating  coil  makes  the  exciting  ampere-turns. 

Autotransformers  may  also  be  used  as  voltage  regulators, 


* Art.  206. 


664 


ALTERNATING  CURRENTS 


as  has  heretofore  been  pointed  out ; * and  a form  of  regulator 
for  single-phase  circuits  was  commonly  used  in  former  years 
which  consists  of  two  coils  placed  at  right  angles  on  the  inner 
barrel  of  a hollow  cylindrical  laminated  core,  one  of  the  coils 
being  connected  across  the  circuit  and  the  other  introduced  in 
series  in  the  feeder  concerned.  A segmented  rotatable  core  is 
used  to  direct  the  magnetic  flux  set  up  by  the  primary  winding 
through  the  secondary  winding  or  shunt  it  therefrom. 

Such  regulators  are  commonly  made  to  vary  the  voltage  of  the 
circuit  to  which  they  are  attached  over  a range  from  10  per  cent 
lowering  to  10  per  cent  boosting.  The  primary  winding  must  be 
designed  for  the  voltage  of  the  main  circuit,  and  the  secondary 
conductors  must  be  capable  of  carring  the  full  feeder  current. 
When  the  regulator  is  raising  the  voltage,  the  outgoing  current 
is  smaller  than  the  current  in  the  main  circuit  leading  to  the 
device,  and  when  the  regulator  is  lowering  the  voltage,  the  out- 
going current  is  larger  than  the  current  in  the  circuit  leading  to 
the  device. 

161.  Connecting  Alternators  for  Combined  Output.  — The  con- 
ditions required  for  successfully  connecting  alternators  so  that 
their  outputs  may  be  combined  are  quite  different  from  those 
obtaining  in  the  case  of  direct-current  machines.  In  order 
that  the  output  of  alternators  may  be  added,  it  is  evident  that 
the  voltage  waves  impressed  by  them  upon  the  circuit  must  be 
in  exact  consonance.  That  is,  the  voltage  waves  must  be  of 
equal  period  or  In  synchronism  and  also  of  corresponding  phase 
or  In  step  with  each  other.  If  this  is  not  the  case,  the  machines 
will  be  in  opposition  during  all  or  a portion  of  the  current  wave. 
The  following  discussions  concerning  combined  output  are 
applicable  to  both  single-phase  and  polyphase  alternators.  In 
case  of  the  latter,  the  voltage  and  current  in  the  windings  of  a 
single  phase  should  be  understood  to  be  referred  to,  while  the 
power  of  the  single  phase  should  be  multiplied  by  the  number 
of  phases  to  get  the  total  machine  power. 

161  a.  Alternators  in  Series.  — The  alternators  will  be  assumed 
in  this  discussion  to  be  constructed  so  as  to  give  equal  currents 
at  equal  frequency.  The  form  of  the  current  waves  will  also 
be  assumed  to  approximate  to  sinusoids  and  the  armatures  to 
have  negligible  reactive  effects  upon  the  field  magnets. 

* Art.  144. 


SYNCHRONOUS  MACHINES 


t>65 


In  Fig.  397  let  the  curves  A and  A'  represent  the  voltage 
waves  measured  respectively  across  the  terminals  of  two  alter- 
nators with  their  armatures  connected  in  series,  the  machines 
being  driven  independently,  but  so  as  to  give  practically  the 
same  frequencies  and 
voltages.  The  ordi- 
nates of  curve  It  are 
the  algebraic  sums  of 
the  corresponding  or- 
dinates of  curves  A and 
A',  and  hence  curve  It 
represents  the  resultant 
voltage  impressed  on 
the  external  circuit  by 
the  two  machines. 

Curve  C is  assumed  to 
be  the  curve  of  lagging 
current  flowing  in  the 
circuit.  Assuming  the 
two  machines  to  be 
running  synchronously, 
but  to  be  out  of  step 
by  an  angle  2 </>,  makes 
<f>  the  phase  difference  between  the  resultant  voltage  wave  and 
either  component  wave.  Finally,  the  current  lags  behind  the 
resultant  voltage  by  an  angle  d,  on  account  of  self-inductance 

in  the  circuit.  The 
work  put  into  the 
circuit  by  each 
machine  is  propor- 
tional to  the  alge- 
braic summation  of 
the  products  of  the 
ordinates  of  the  re- 
spective  voltage 

Fig.  398.  — Power  Loops  of  Alternators  producing  the  Vol-  wave  with  the  CU1'- 
tages  and  Current  shown  in  Fig.  397.  , rT, , 

° s rent  wave.  The 

total  work  done  in  the  circuit  is  equal  to  the  sum  of  the  prod- 
ucts of  the  ordinates  of  the  current  and  resultant  voltage  curves. 
Therefore,  since  the  voltage  wave  of  the  lagging  machine  is 


Fig.  397.  — Curves  of  Voltages  and  Current  show- 
ing Unstable  Conditions  existing  when  Alternators 
are  Connected  in  Series. 


666 


ALTERNATING  CURRENTS 


nearest  the  current  wave,  that  machine  furnishes  more  work  to 
the  circuit  than  does  the  leading  machine.  The  power  loops 
for  the  two  machines  are  shown  by  the  curves  a and  a'  in  Fig. 
398.  The  power  delivered  to  the  circuit  by  one  machine  is 

represented  by  the 
height  of  the  line  xx 
above  the  JT-axis,  and 
the  power  delivered  to 
the  circuit  by  the  other 
machine  is  represented 
by  the  height  of  the 
line  x'x'  above  the  axis. 

If  the  two  machines 
were  rigidly  connected 
together,  this  condition 
would  continue  indefi- 
nitely. Assuming,  how- 
ever, that  the  machines 
are  driven  by  separate 
t . „ , ,,  u , „ . , belts,  or  attached  to 

Fig.  399.  — Curves  of  voltages  and  Current  snowing  ’ 

Unstable  Conditions  when  Alternators  are  con-  separate  engines,  the 
nected  in  Series.  lag'crinsf  machine  be- 

OO  O 

ing  the  more  heavily  loaded,  tends  to  fall  farther  behind  its 
more  lightly  loaded  mate,  and  a still  greater  percentage  of  the 
load  is  thrown  upon  it.  At  the  same  time,  as  is  illustrated  by 
Fig.  399,  this  re- 
duces the  total 
work  done  in  the 
external  circuit,  for 
the  total  voltage 
wave  is  now  of  less 
height  than  it  was 
when  the  c o m - 
ponent  curves  were 
nearer 

of  phase.  The 
power  loops  for  the  condition  of  Fig.  399  are  shown  in  Fig.  400. 
The  height  of  the  line  xx  has  decreased,  and  that  of  x' x'  has 
increased,  but  the  sum  of  the  heights  is  less  than  before.  The 
tendency  of  the  lagging  machine  to  fall  farther  behind  may 


a a 

coincidence  Fig.  400.  — Power  Loops  of  Alternators  producing 
Voltages  and  Current  shown  in  Fig.  399. 


the 


SYNCHRONOUS  MACHINES 


667 


Fig.  401.  — Voltages  of  Two  Alternators  connected 
in  Series,  after  they  have  the  Stable  Position  of 
Series  Opposition. 


continue  until  the  voltage  waves  of  the  two  machines  approach 
exact  opposition  to  each  other  in  the  series  circuit  (Fig.  401). 
When  in  opposition  the  machines  are  in  stable  equilibrium,  but 
are  delivering  no  power  to  the  external  circuit. 

If  the  machines  were  started  with  their  voltage  waves  in  exact 
step,  they  would  do  equal  work,  but  their  equilibrium  would  be 
unstable,  and  any  disturbance  of  their  relations  would  cause  them 
to  tend  towards  opposition.  It  is  therefore  not  possible  to  operate 
alternators  in  cumula- 
tive series  on  an  induc- 
tive circuit  unless 
they  are  rigidly  united 
by  a mechanical  coup- 
ling. This  result  also 
follows  when  the  nor- 
mal voltages  of  the 
machines  are  different, 
in  which  case  the  vol- 
tage impressed  on  the 
circuit  when  equilibrium  is  attained  is  the  difference  of  the 
machine  voltages.  Even  if  there  were  no  reactance  in  the  ex- 
ternal circuit  on  which  the  machines  were  working,  the  resultant 
voltage  and  current  would  have  the  same  phase,  and  the  ma- 
chines would  be  in  equilibrium;  but  the  equilibrium  would  be 
unstable,  for  after  any  disturbance  of  the  operation  of  the 
machines  they  would  have  no  tendency  to  return  to  their 
former  operating  state. 

The  condition  of  operation  of  two  alternators  connected  in 
series  on  an  inductive  circuit  is  also  plainly  indicated  by  means 
of  a vector  diagram  (Fig.  402).  In  this  diagram  the  lines 
represent  quantities  as  follows: 

Voltage  of  leading  machine  = OA. 

Voltage  of  lagging  machine  = OA'. 

Resultant  voltage  in  circuit  = OR. 

Current  in  circuit  = OC. 

Power  given  to  circuit  by  first  machine  = Oa  x OC. 

Power  given  to  circuit  by  second  machine  = Oa’  x OC. 

Total  power  given  to  circuit  = Or  x OC  = (Oa  + Oa ')  x OC. 

It  is  evident  from  the  construction  that  if  the  angle  ROC  is 
fixed  by  the  conditions  of  the  circuit,  then  the  length  of  OR  de- 


668 


ALTERNATING  CURRENTS 


R 


Fig.  402.  — Vector  Diagram  of  Voltages  and 
Currents  for  Two  Alternators  in  the  Unstable 
Condition  of  Series  Operation. 


creases  as  the  angle  2 <p  increases,  and  that  Oa  decreases  at  the 
same  time;  but  Oa!  increases  for  a time  and  then  decreases  at  a 
less  rate  than  Oa , so  that  the  machines  tend  to  get  farther 
apart  in  phase.  When  = 90°  — 0,  the  length  of  Oa  vanishes, 

the  first  machine  gives  no 
power  to  the  circuit,  and 
all  the  power  is  furnished 
by  the  second  machine. 
When  cf)  approaches  more 
nearly  90°,  or  exceeds  that 
angle,  as  when  the  machines 
are  approaching  opposi- 
tion, one  machine  will  run 
as  a motor  if  its  prime  mover  is  disconnected.  Of  course,  when 
OR  decreases,  if  the  resistance  of  the  circuit  is  unaltered,  the 
current  OC  also  decreases,  but  the  relative  outputs  and  phases 
of  the  machines  are  not  altered  thereby. 

In  case  capacity  reactance  could  be  placed  in  the  series 
circuit  between  the  machines  sufficient  to  cause  the  current  to 
lead  the  resultant  voltage  by  an  angle  d,  the  condition  is 
represented  by  Fig.  403,  in  which  the  vectors  are  lettered  as 
in  Fig.  402.  In  this  case, 

Oa  is  greater  than  Oa' ; 
that  is,  the  leading  machine 
furnishes  the  greater 
amount  of  power,  and  the 
machines  tend  to  come 
together  and  run  in  cumu- 
lative series.  Similar  de- 
ductions can  be  drawn  from 
diagrams  like  Figs.  397  and 
398,  but  with  the  current  wave  in  the  lead  of  the  resultant  voltage 
wave.  Since  the  alternator  windings  always  contain  self-induct- 
ance, special  insertions  of  capacity  in  the  series  circuit  would 
have  to  be  made  to  get  sufficient  capacity  reactance  to  produce 
this  result,  and  it  may  therefore  be  said  that,  in  general,  alter- 
nators in  series  are  not  stable  in  operating. 

161  b.  Alternators  in  Parallel.  — When  the  machines  have 
reached  series  opposition  of  phases,  as  explained  above,  their 
voltages  are  in  the  proper  relation  to  cause  them  to  work  in 


Fig.  403. — Vector  Diagram  of  Voltages  and 
Current  of  Two  Alternators  in  Series  when 
working  on  a Load  of  High  Capacity  Reactance. 


SYNCHRONOUS  MACHINES 


669 


parallel  in  delivering  current  to  an  external  circuit  attached  be- 
tween the  lead  wires  connecting  their  terminals;  for  it  will  be 
seen  by  reference  to  Figs.  401  and  404  that  when  machines  A 
and  A'  are  in  series  opposition  to  each  other,  the  points  R and 
S must  at  every  instant  be  of  opposite  sign,  and  that,  therefore, 
the  machines  will  deliver  current  through  the  load  circuit  m, 
m , m.  This  is  also  the  proper  connection  for  direct  current 
generators  arranged  for  parallel  operation  ; that  is,  the  voltages 
are  opposed  to  each  other  in  the  series  circuit  through  both 
armatures. 


- R m 


Fig.  401.  — Diagram  of  Two  Alternators  connected  for  Parallel  Operation. 


The  operation  of  alternators  in  parallel  was  first  achieved  by 
Wilde  in  1868,*  but  this  work  was  overlooked  during  the  period 
of  development  of  the  direct-current  dynamo.  In  1884  Dr.  John 
Hopkinson  showed  by  mathematical  analysis  the  practicability 
of  working  them  in  parallel.  This  was  done  without  a knowl- 
edge of  Wilde’s  earlier  experiments,  and  it  led  to  some  experi- 
ments which  were  carried  out  by  Hopkinson  and  Adams  upon 
De  Meritens  magneto  machines,  f These  experiments  fully 
bore  out  Hopkinson’s  deductions,  but  their  practical  bearing 
was  not  fully  appreciated  until  a few  years  later,  when  the 
transformer  system  of  alternating-current  distribution  was  de- 
veloped. 

161  c.  Synchronizing  Current. — Successful  parallel  operation  of 
alternators  depends  upon  their  holding  each  other  in  synchronism 
and  step  (by  the  motor  action  referred  to  later  in  the  article), 
even  when  the  prime  movers  do  not  naturally  synchronize. 
The  effort  of  the  machines  to  do  this  causes  the  flow  of  a cross 
current  between  them,  which  is  commonly  called  the  synchro- 
nizing current.  This  component  of  the  total  current  from  a 
machine  may  be  considered  as  a series  current  circulating  be- 

* See  Philosophical  Magazine , Vol.  37,  4th  series,  1869,  p.  54. 
t Adams,  Jour.  Inst.  E.  E.,  Vol.  13,  1884,  p.  515. 


670 


ALTERNATING  CURRENTS 


tween  the  machines  considered  as  pairs  connected  to  the  station 
bus  bars.  The  remaining  component  of  the  current  from  each 
machine  joins  in  parallel  with  that  of  the  other  machines  and 
flows  through  the  load.  The  total  synchronizing  current  flow- 
ing between  a machine  and  the  station  bus  bars  may  be  re- 
solved into  two  components,  one  in  phase  with  the  voltage  which 
causes  the  synchronizing  current  to  flow,  and  the  other  lagging 
90°  behind  the  voltage.  The  former  is  usually  quite  small. 
Thus,  suppose  in  Fig.  405  that  Oa  and  Ob  represent  equal 
vector  voltages  of  two  generators  A and  B , taken  in  series  cir- 
cuit through  the  two  armatures.  The  machines  may  be  con- 
sidered as  connected  to  the  same  bus  bars  of  a central  station. 
Curves  a and  b are  sinusoids  representing  the  voltage  waves. 
Assume  for  the  present  that  the  machines  are  carrying  no  ex- 
ternal load  and  that  the  prime  mover  of  B is  not  supplying 
quite  enough  power  to  drive  it  unloaded  while  that  of  A is  not 
only  able  to  drive  A but  also  to  aid  B.  Then  machine  B will 
tend  to  fall  back  in  phase  or  step  and  its  voltage  curve  b will 
lag  with  reference  to  the  voltage  curve  a of  the  machine  A. 
As  will  be  seen  this  will  tend  to  cause  a motoring  synchronizing 
current  to  flow  through  the  series  circuit  comprising  the  arma- 
tures of  the  machines.  Thus,  since  a and  b are  slightly  out  of 
opposition,  there  will  be  a resultant  voltage  Oq  (curve  g),  tend- 
ing to  set  up  a series  synchronizing  current.  The  relations  are 
simply  shown  in  the  vector  diagram  in  the  figure,  where  the 
outer  ends  of  the  lines  representing  the  vector  voltages  and 
currents  corresponding  to  the  curves  are  marked  with  the  same 
letters  as  the  respective  curves.  The  series  current  set  up  by 
voltage  q may  be  considered  as  made  up  of  a component  s,  in 
phase  with  q , and  of  another  component  u , in  quadrature  with 
q , the  former  being  the  active  and  the  latter  the  wattless  com- 
ponent. The  component  s has  the  same  effect  upon  both  a and 
b during  a complete  period,  i.e.  each  machine  furnishes  one  half 
the  power  required  to  drive  s through  the  series  circuit,  hence  it 
can  have  no  effect  in  tending  to  draw  the  trailing  machine  into 
the  proper  phase  relation  with  its  mate;  but  the  component  u 
(90°  from  j),  which  is  dependent  upon  the  self-inductance  of 
the  series  circuit,  must,  from  its  position,  cause  an  accelerating 
torque  on  the  lagging  machine  (5),  and  a corresponding  re- 
tarding torque  on  the  leading  machine;  and  it  thus  tends  to 


SYNCHRONOUS  MACHINES 


671 


draw  the  machines  into  synchronism  and  step  for  proper  parr 
allel  operation.  It  will  be  noticed  by  reference  to  the  figure 
that  u is  in  phase  with  m and  in  opposition  to  and  that  m and 


a 

o 

S-l 

aa 

o 

a 

rJl 

c3 


a 

o — • 
% © 


ps  « 


a 

o 

bJD 

£ 


n are  respectively  the  components  of  a and  b , which  are  in  op- 
position to  each  other  in  the  series  circuit,  and  are  therefore 
impressed  on  the  parallel  circuit  ; also  that  the  motoring  effect 


672 


ALTERNATING  CURRENTS 


on  B is  proportional  to  Ou  times  On , while  the  retarding  effect 
on  A is  proportional  to  Ou  times  Om.  If  the  machine  B led  the 
machine  A , then  the  synchronizing  current  would  retard  instead 
of  assisting  the  former,  as  the  figure  plainly  shows. 

The  effect  of  the  synchronizing  current  in  dragging  the  ma- 
chines into  step  depends  upon  its  magnitude  and  relative  phase  ; 
while  its  magnitude  at  each  instant  depends  : (1)  upon  the 
algebraic  sum,  or,  what  is  the  same  thing,  the  arithmetical  dif- 
ference between  the  instantaneous  voltages  developed  by  the 
machines ; (2)  upon  the  reciprocal  of  the  impedance  of  the 
machine  armatures.  In  other  words,  the  instantaneous  synchro- 
nizing current  flowing  in  series  relation  through  the  two  arma- 
tures, if  the  two  machines  are  alike,  is 

ia  = e°  + e*> 

2 V B *+  4 t t*PL? 

where  ea  and  eb  are  the  instantaneous  voltages  of  the  machines 
which  are  of  ojiposite  sign  and  equal  when  the  machines  are  in 
exact  synchronism  and  step,  and  Ra  and  La  are  respectively 
the  resistance  and  the  inductance  of  the  armature  circuit  of 
each  machine  including  the  leads  from  the  bus  bars.  It  is 
here  assumed  that  the  effective  voltages  of  the  two  machines 
are  numerically  equal,  which  is  an  essential  condition  for  the 
synchronizing  current  to  be  practically  wattless,  and  is  the  con- 
dition in  which  it  is  aimed  to  run  alternators  in  practice. 

Now  suppose  the  voltage  curve  of  the  machine  B to  lag  be- 
hind the  phase  of  the  voltage  curve  at  the  bus  bars  by  an  angle 
fi.  Let  the  effective  values  of  these  voltages  be  E.  and  the 
corresponding  maximum  voltage  be  em.  At  any  moment  the 
instantaneous  voltages  are  ea  and  e6,  and 

ea  = em  sin  « = v/2  E sin  a, 

eb  = em  sin  («  + 180°  — /3)  = V2  E sin  («  -f  180°  — $), 

when  considered  to  be  acting  in  the  series  circuit.  The  in- 
stantaneous voltage  causing  a synchronizing  current  to  flow  is 
the  algebraic  sum  of  these,  or 

ea  + eb  — V2  E [sin  a + sin  (a  + 180°  — /3)]. 

It  is  evident  from  Fig.  405  that  this  is  a maximum  at  the  in- 
stant when  ea  and  eb  are  equal  and  of  similar  signs,  in  which 


SYNCHRONOUS  MACHINES 


673 


case  a = .f  /3.  Tlien  the  maximum  value  of  the  voltage  causing 
a synchronizing  current  is  found  by  substituting  this  value  of  a 
in  the  expression  for  ea  + eb  and  ( ea  + eb)m  •=  2V2i?sin  /3 . 

The  effective  value  of  the  voltage  is  then  2 E sin  | /3,  and 
the  synchronizing  current  is 

I _ E sin  j,  /3 


For  smooth  and  successful  working  in  parallel  Is  must  become 
sufficiently  great,  in  case  the  machines  tend  to  get  out  of  step, 
to  pull  the  machines  together  before  /3  becomes  of  appreciable 
magnitude.  Hence  it  is  desirable  that  the  denominator  in  the 
expression  for  Is  be  reasonably  small.  In  other  words,  arma- 
ture impedance  must  be  small  to  give  smooth  parallel  work- 
ing. Figure  405  shows  plainly  that  the  voltage  ( Oq ) causing 
I3  is  behind  the  phase  of  the  voltage  of  the  trailing  machine 
by  an  angle  90°  — 1 /3.  The  synchronizing  current  (Z,  = Oc ) 
lags  behind  the  voltage  Oq  by  an  angle  9S  (angle  qOc ),  the 


tangent  of  which  is 


2tt  fLa 

Rn 


Consequently  the  synchronizing 


current  is  out  of  phase  with  the  trailing  machine  voltage  by  an 
angle  of  90°  + ( 9a  — 1/3)  and  out  of  phase  with  the  leading 
machine  voltage  by  an  angle  of  90°  — (08 -f  | /3).  A current 
(wattless)  which  has  a phase  difference  of  180°  or  0°  compared 
with  the  voltage  of  a machine  has  the  strongest  effect  in  bring- 
ing the  machine  into  step  by  tending  to  accelerate  it  or  to  retard 
it.  This  effect  is  to  accelerate  or  retard  the  refractory  machine 
depending  on  whether  the  phase  difference  is  180°  or  0°,  which 
in  turn  depends  upon  whether  the  machine  is  trailing  or  leading. 

The  component  Ou(  = 7^)  of  the  synchronizing  current  just 
found  is 


and  therefore, 


I = 


1 * = la  Sin 

E sin  i f3  2 i rfLa 

V^2  + 4 ri2f2La2  vX2  + 4 7 T2pL2 


or. 


-^  = E sin  -t  /3 


2 7rfLa 

Ra  + 4 7 r2PL2 


and,  with  a fixed  value  of  Ra,  this  will  have  a maximum  value 
when 

Ra=  2 7 rfLa  or  when  tan  0S  = 1 and  0S  = 45°. 

2 x 


674 


ALTERNATING  CURRENTS 


The  maximum  possible  value  of  Iu  where  the  machine  cir- 
cuit has  a given  resistance  is  therefore 


1^  max  = 


E sin  t /3 
2 Ra  ’ 


and  the  corresponding  value  of  I„  the  total  synchronizing  cur- 
rent, is 

T _ Esin  1/3 
-*~8  — 

V2  Ra 


The  limits  in  the  value  of  Ra  are  fixed  by  considerations  of 
economy  in  construction  and  of  efficiency,  and  the  frequency 
is  fixed  by  conditions  of  operation  ; La  is  therefore  the  only 
independent  variable  in  the  preceding  equations.  In  order  to 
have  the  most  sensitive  mutual  control,  the  self-inductance  of 
the  armature  circuit,  which  at  its  least  value  is  always  many 


R 

times  larger  than  - ° ■ , must  be  as  small  as  commercially  pos- 

2 7 TJ 

sible.  If  it  were  possible  to  reduce  the  reactance  to  the  value 
of  the  resistance,  the  jerking  of  a refractory  alternator  into 
phase  would  probably  be  too  severe  for  good  working,  and  the 
stresses  imposed  on  the  machine  might  be  injurious;  but  in 
most  commercial  machines  such  trouble  is  not  likely  to  exist  on 
account  of  the  unavoidable  magnitude  of  the  self-inductance, 
which  cannot  be  reduced  beyond  certain  limits. 

The  foregoing  equations  are  developed  on  the  hypothesis  of 
two  equal  alternators  operated  in  parallel,  but  the  conclusions  are 
equally  applicable  to  two  alternators  of  unequal  internal  resist- 
ances and  reactances  and  also  to  the  conditions  of  many  alter- 
nators connected  in  parallel  to  the  same  bus  bars.  In  either 
case  the  impedance  of  the  cross  current  circuit  for  any  particular 
machine  is  equal  to  the  sum  of  the  impedances  of  the  arma- 
ture and  connecting  leads  of  that  machine  in  series  with  the 
circuit  composed  of  its  mates  in  parallel  and  their  connections. 
The  bus  bar  voltage  may  be  considered  to  be  unaffected  by  the 
cross-current  flow ; then  the  equations  for  is  and  Is  show  that 
their  values  are  as  large  as  the  values  given  in  the  foregoing 
paragraphs,  for  the  reason  that  the  resultant  series  voltage  Oq 
for  one  machine  only  acts  in  a circuit  of  impedance  made  up  of 
the  armature  and  leads  of  only  one  machine. 


SYNCHRONOUS  MACHINES 


675 


Fig.  406.  — Vector  Dia- 
gram of  Voltages  and 
Synchronizing  Current 
when  One  Alternator 
in  Parallel  with  Others 
lags  in  Step. 


The  vector  diagrams,  Figs.  406  and  407,  contrast  the  con 
ditions  of  a leading  and  a trailing  alternator  with  respect  to 
bus  bars  of  fixed  voltage  OA.  Figure  406 
shows  the  position  of  the  voltage  OB  of  the 
trailing  alternator  and  the  position  of  the 
synchronizing  current  Oo  which  tends  to 
accelerate  the  alternator  into  step ; while 
Fig.  407  shows  the  corresponding  conditions 
for  an  alternator  with  leading  voltage  OB 
and  synchronizing  current  Oc  which  tends 
to  retard  the  alternator  into  step. 

In  each’ of  these  figures, 

OA  represents  the  bus  bar  voltage  in 
phase  and  magnitude. 

OB  represents  the  machine  voltage  in 
phase  and  magnitude. 

OQ  represents  the  resultant,  or  synchro- 
nizing, voltage  in  phase  and  magnitude. 

Oc  represents  the  synchronizing  current 
in  phase  and  magnitude. 

A Ou  represents  its  wattless,  or  phasing, 

component. 

/3  is  the  angle  by  which  the  machine 
differs  from  its  proper  phase  for  parallel 
working,  OB1. 

6,  is  the  angle  QOc  by  which  the  synchro- 
nizing current  lags  behind  the  resultant 
voltage. 

The  figures  show  plainly  that  if  OA  and 
OB  remain  constant,  then  as  /3  increases  OQ 
increases  and  at  the  same  time  Oc  and  Ou 
increase.  The  product  OB  x Ou  x cos  \ /3 
is  proportional  to  the  synchronizing  torque 
exerted  on  the  machine.  It  is  negative  in 
Fig.  406  and  positive  in  Fig.  407,  so  that 
hen  One  Alternator  the  torque  is  exerted  on  the  machine  and 
m Parallel  with  others  accelerates  it  in  one  case,  and  it  is  exerted 
step.  by  £pe  macpine  auq  retards  it  in  the  other 

case.  For  the  conditions  of  OB  = OA  and  a given  value  of  /3,  the 
torque  is  a maximum  when  Ou  x OB  is  a maximum,  which 


Fig.  407.  — Vector  Dia- 
gram of  Voltages  and 
Synchronizing  Current 


676 


Alternating  currents 


occurs  when  2 n/L  — E,  or  6S  — 45°.  In  a 2200-volt  500-kilo- 
watt  machine  giving  a frequency  of  60  periods  per  second  and 
having  a resistance  in  the  armature  circuit  of  .5  ohm,  it  is 
required  that  L = .0013  to  give  a maximum  synchronizing 
effect,  which  is  smaller  than  the  smallest  value  which  is  now 
commercially  attainable  in  any  type  of  alternator  of  that  size. 

An  inspection  of  the  figures  shows  that  if  the  synchronizing 
current  Oc  led  the  resultant  voltage  OQ , there  would  be  no 
tendency  for  the  machine  to  come  into  parallel  step  with  the  bus 
bar  voltage ; but  this  condition  will  not  occur  fn  practice,  since 
it  is  impossible  to  build  alternators  without  self-inductance  in 
the  windings. 

In  the  foregoing  discussion,  the  usual  condition  in  which  a 
prime  mover  tends  to  lag  or  race  in  speed  is  considered ; and 
it  is  shown  how  a synchronizing  cross  current  is  set  up  through 
alternators  operating  under  such  conditions,  which  current 
tends  to  hold  the  generators  at  speeds  giving  a common  fre- 
quency for  the  voltage.  This  current  causes  a transfer  of 
power  from  a leading  machine  to  its  mates  or  from  the  mates 
to  a trailing  machine.  The  current  is  therefore  wattless  with 
respect  to  the  system,  except  for  PR  losses. 

162.  Division  of  Load  between  Parallel  Alternators.  — Alter- 
nators may  be  completely  synchronized  but  yet  fail  to  properly 
divide  the  load  of  the  external  circuit  between  them,  as  the 
driving  torque  of  the  prime  mover  of  each  machine  may  not 
be  adjusted  to  exactly  overcome  at  the  correct  speed  the  resist- 
ing moment  of  the  load  which  that  machine  should  carry.  The 
condition  is  illustrated  in  Fig.  408,  where  the  conditions  are  the 
same  as  assumed  in  the  preceding  article  except  that  the  ex- 
ternal circuit  is  receiving  power.  In  this  figure,  OT \ is  the  bus 
bar  voltage  E which  is  assumed  to  be  constant  in  length  and  is 
used  as  the  reference  phase  position  with  respect  to  time.  The 
vector  of  current  I in  the  external  circuit  is  represented  by  the 
line  OH , which  is  shown  lagging  behind  the  line  voltage  by  an 
angle  6 , and  which  is  dependent  for  its  value  upon  the  char- 
acter and  value  of  the  impedance  of  the  external  circuit. 

Assume  first  that  two  exactly  equal  machines  are  connected 
to  the  bus  bars  and  are  excited  so  as  to  give  equal  total  vol- 
tages. Further  assume  that  the  governors  of  the  driving  en- 
gines are  so  adjusted  as  to  give  exactly  equal  torques.  Then 


SYNCHRONOUS  MACHINES 


nrrrr 

bt  t 


the  voltage  of  the  generators  must  be  in  exact  opposition  of 
phase  so  far  as  the  series  circuit  through  the  two  armatures 
are  concerned,  and  no 
series  current  will  flow,  but 
they  are  in  exact  parallel 
relation  with  reference  to 
the  external  circuit. 

These  relations,  under 
these  circumstances,  are 
quite  similar  to  that  of  two 
equal  direct-current  gen- 
erators delivering  equal 
loads  to  an  external  circuit. 

The  value  and  phase 
position  of  the  alternator 
voltage  required  to  furnish 
the  external  power,  P — 

El  cos  d,  may  be  obtained 
as  follows : lay  out  TM 
parallel  to  OH  and  of  such 
a length  that  it  equals 
RaI,  where  Ra  is  the  re- 
sistance of  one  of  the 
armature  windings.  At 
right  angles  to  TM  lay  off 
MO  of  a length  equal  to 
XaI,  where  Xa  is  the 
synchronous  reactance  of 
one  of  the  armature  wind- 
ings. Then  CT=IZa  = Ea 
represents  the  voltage  drop 
in  the  parallel  impedances 
of  the  two  armatures  when  current  I flows.  The  generated 
armature  voltages  must  then  be  vectorially  equal  to  OT  plus 
TC ',  or  Eg  = E + Ea.  When  considered  with  reference  to  the 
series  armature  circuit  the  total  generated  voltages  may  be  laid 
off  as  00  and  OC,  as  was  done  in  Figs.  405,  406,  and  407. 
The  load  given  out  by  each  machine  is, 
ssrn  TM  1 


Fig.  108.  — Vector  Diagram  for  showing  the 
Division  of  Current  and  Load  between  Two 
Equal,  equally  Excited  Alternators  connected 
in  Parallel. 


678 


ALTERNATING  CURRENTS 


since 


TM=  HRJi  IRa  and  \ I = T^-, 

tin 


and  the  total  power  generated  by  each  machine  is 

i p — i p _i_  i nj? 

Suppose,  now,  that  the  governors  of  the  driving  engines  are 
so  adjusted  that  one  has  a greater  torque  than  the  other,  but 
so  that  the  external  load  is  not  changed.  The  two  machine 
voltages  will  then  fall  apart  in  phase,  say  by  the  angle  /3  (Fig. 
408).  Suppose  also  that  the  total  voltages  of  the  two  genera- 
tors are  again  equal  in  scalar  value.  Construct  the  isosceles 
triangle  OAB  so  that  the  base  AB  passes  through  the  point  C 
and  is  at  right  angles  to  CC,  and  the  sides  OA  and  OB  make 
the  required  angle  /3.  OA  and  OB  are  the  required  generator 
voltages  and  combine  in  parallel  to  send  the  load  current  I — 
OH  through  the  circuit  composed  of  the  line  in  series  with  the 
two  generator  armatures  in  parallel.  The  resultant  AB  being 
at  right  angles  with  the  voltage  OC  it  has  no  effect  upon  the 
external  current  OH.  Now  draw  the  lines  TA  and  TB  and 
erect  the  internal  armature  voltage  triangles  TNA  and  TSB. 
The  current  generated  by  machine  A is  proportional  to  the 

active  side  of  TNA , or  TN  or  IA  — when  ER  is  the 

Ba  Ra 

active  voltage  lost  in  the  generator  copper.  Likewise,  the 
total  current  flowing  in  generator  B is  proportional  to  TS  or 

IB  = As  the  generators  are  similar  and  equal,  the 

b a B a 

X 

ratio  — - is  the  same  for  both  and  for  the  two  in  parallel. 
Ba 

Scale  TN  and  TS  in  terms  of  currents  and  then  we  have  their 
resultant  TK  equal  to  the  total  line  current  OH  or  double  TOL 
i.e.  1=  IA  + IB ; while  their  vector  difference  equals  SN  and 

The  total  power  gen- 


. , . r SN  — IB 

the  series  current  is  Js  = — = — - 


erated  by  machine  A is  the  product  of  the  projection  of  TN  on 
OA  with  OA  and  of  machine  B the  product  of  the  projection  of 
TS  on  OB  with  OB. 

For  indicating  the  series  relation,  one  of  the  generator  vol- 
tage vectors  can  be  conveniently  reversed.  Thus,  we  obtain 


SYNCHRONOUS  MACHINES 


679 


the  parallelogram  OAFB'  which  is  similar  to  those  of  Figs. 

405,  406,  and  407.  The  resultant  series  voltage  OF  equals 

BA , while  the  series  current  which  would  flow  because  of  OF 

. OG  , . , . , , SN 

is  , which  is  equal  to  . 


If  OA  and  OB  are  not  equal,  the  triangle  OAB  is  no  longer 
isosceles,  but  the  construction  can  be  carried  out  in  the  same 
manner  as  above. 

However,  instead  Q 

of  going  further 
into  this  special 
case  of  two  alter- 
nators, consider 
the  more  general 
case  of  one  alter- 
nator connected  to 
bus  bars  to  which 
are  connected  a 
large  capacity  of 
other  alternators, 
so  that  the  bus  bar 
or  external  voltage 
remains  practically 
constant  in  value. 

In  order  to  find  the 
current  and  power 
that  will  be  pro- 
duced by  a machine 
having  an  induced 
voltage  Eg  of  a 
given  scalar  value 
equal  to  the  radius 
of  circle  MM  with 
center  at  0,  but  of 
variable  phase  po- 
sition, the  follow- 
ing construction 
can  be  made;  lay  out  OF,  as  in  Fig.  409,  equal  to  the  bus 
bar  voltage  E,  and  assume  that  the  driving  engine  torque 
is  such  that  the  generated  voltage  Eg  makes  angle  (3  with  E. 


Fig.  409.  — Diagram  showing  Locus  of  Current  for  Con- 
stant Bus  Bar  Voltage  and  Generator  Voltage. 


680 


ALTERNATING  CURRENTS 


Then  OB=Eg.  The  internal  voltage  is  then  B V.  Lay  out  on 
BV  the  internal  voltage  triangle  BVN,  in  which  BV=IZa 
is  the  total  loss  of  voltage  internally,  and  is  composed  of 
the  active  and  reactive  voltages  VN  = IRa  and  NB  = IXa , 
respectively.  In  this  case,  as  in  those  preceding,  synchro- 
nous impedance  and  reactance  are  used.  The  apparent  vol- 

E 

tage  or  impedance  triangle  in  which  Eg  or  — ? is  the  hypot- 
enuse is  OHB.  The  apparent  impedances  in  this  case  are 
composed  of  the  self-inductances,  field  reactions,  voltage  re- 
actions, and  resistances  encountered  throughout  the  system 
in  absorbing  Eg.  With  any  change  of  the  driving  engine 
torque,  which  changes  ft,  the  form  of  the  triangle  OHB 
changes.  In  the  figure  with  /3  = angle  V OB  the  generator 
current  leads  the  bus  bar  voltage  by  the  angle  JVN,  or  6 is 
negative. 

The  side  OH,  however,  must  always  be  parallel  to  VN,  the 
active  voltage  line,  and  the  angle  OHB  is  always  a right  angle. 

In  finding  the  current  locus  as  f3  varies  or  B moves  along  its 
circular  locus  MM,  the  following  symbols  are  used  : 


The  generated  voltage,  OB  = Eg. 

The  bus  bar  voltage,  0 V = E. 

The  angle  between  Eg  and  E,  VOB  — /3. 

The  internal  voltage,  BV—  Ez. 

The  internal  active  voltage,  VN  = ER. 

The  internal  reactive  voltage,  NB  — Ex. 

The  armature  current  scaled  to  equal  VN  = I. 


The  internal  angle  of  lag, 


NVB  = 6„ 


The  angle  between  I and  E,  NVI  = 6. 

The  armature  resistance  = Ra. 

The  synchronous  armature  reactance  = Xa. 

The  synchronous  armature  impedance  = Za. 

The  locus  of  the  point  B is  determined  by  the  formula  for 
the  triangle  OBV, 


Eg 2 = EJ  - 2 EZE  cos  [180°  - (0a  - <?)]  + E\ 


where  6 is  intrinsically  positive  or  negative  depending  upon 
whether  it  is  to  the  right  or  left  of  OJ ; in  the  figure  for  point 
B it  is  negative.  When  E and  Ee  are  of  constant  values  and 


SYNCHRONOUS  MACHINES 


681 


6 is  variable,  this  is  also  the  formula  in  polar  coordinates  of  the 
circular  arc  MM  having  the  radius  Eg  and  the  pole  at  V.  To 
find  the  locus  of  the  current  when  6 varies,  the  above  formula 
may  be  modified  by  multiplying  both  sides  by  cos2#a,  which 
gives 

Eg2  cos2  0a=  ER2—  2 ER(E  cos  da)cos  [180°—  (9a—9)~\  +E2  cos2  0a. 

This  is  evidently  the  formula  in  polar  coordinates  of  a circle 
having  a radius  Eg  cos  9a  and  in  which  the  initial  line  is  on  a 
diameter ; also  one  of  the  polar  coordinates  ER , or  taken  to 
E 

proper  scale  it  is  which  equals  i,  the  current  flowing  in 

R 

the  armature.  Now,  swinging  Eg  over  to  the  position  OJ , the 
triangle  of  internal  voltages  becomes  VSJ,  the  angle  JVS 
being  equal  to  6a.  When  Eg  is  thus  in  phase  with  E , the  in- 
ternal drop  Ez  — VJ  and  equals  Eg  — E.  Therefore,  under 
those  conditions  the  active  voltage  is 

Ex'=  VS  = {Eg  — E')  cos  0a. 

Likewise,  when  Eg  is  swung  around  to  OJ' , the  active  voltage 
is 

Er"=  VT=(Eg  + E)cos0a. 

By  this  construction  TVS  is  a straight  line  and  its  value  is 
TS  = ERr  + ER"=  2 Eg  cos  6a. 

One  half  of  this  or  Eg  cos  9a  is  the  radius  of  the  circular  locus 
for  Er  given  above.  Therefore,  bisect  the  line  TS , as  shown 
at  O',  and  describe  the  side  TNS  by  the  radius  O'  T=Eg  cos  6a. 
When  Eg  is  at  the  position  OJ  the  second  coordinate  is  O'V, 
and 

O'  V=  O'S-  VS=  Eg  cos  0a  - [Eg  - E]  cos  0a, 
or  O'  V — E cos  0a, 

which  is  the  value  given  in  the  formula.  V is  evidently  the 
origin  and  TS  the  initial  line  for  the  circle  TNS  as  given  by 
the  formula.  For  the  pair  of  coordinates  VS  and  VO'  the 
angle  included  is  180°,  which  is  correct  as  180 ° —(0a  — 0)  is 
180°  when  Eg  — OJ.  The  circle  TNS  is  then  the  locus  of  the 
active  internal  voltage  ER,  which  has  its  pole  at  V and  its  initial 


682 


ALTERNATING  CURRENTS 


E 

line  TS ; or,  if  the  scale  is  changed  by  a ratio  equal  to  it  is 
the  locus  of  the  armature  current. 

The  power  given  out  is 

P = El  cos  0, 

where  0 is  the  phase  angle  between  E and  I or  is  NVJ  when  Eg 
is  at  the  position  OB.  The  maximum  power  is  reached  when 
Eg  has  swung  forward  so  that  I takes  the  position  VQ,  where  Q 
is  the  point  of  tangency  of  the  current  locus  to  a perpendicular 
to  OV.  At  this  point  Eg  has  swung  to  OB' , where  /3—0a.  If 
the  driving  torque  is  great  enough  to  drive  the  armature  farther 
ahead,  the  machine  will  break  from  synchronism.  The  minimum 
power  is  found  when  Eg  has  fallen  back  so  that  I lies  at  the 
position  VL,  which  is  perpendicular  to  OV  = E.  If  the  torque 
of  the  driving  engine  is  reduced  so  that  Eg  falls  still  farther 
back,  current  is  driven  through  the  armature  by  the  voltage  of 
the  bus  bars  and  the  machine  runs  as  a synchronous  motor. 

Heretofore  we  have  dealt  with  a machine  in  which  the  in- 
duced voltage  was  kept  constant.  An  equally  important  con- 
sideration is  the  effect  of  the  variation  of  this  voltage.  Assume 
that  the  load,  including  the  internal  copper  losses,  and  the  bus 
bar  voltage  are  kept  constant,  but  that  the  generator  excitation 
is  varied.  In  order  that  the  load  may  be  kept  constant,  the 
torque  of  the  driving  engine,  as  determined  by  the  setting  of 
its  governor,  is  kept  constant.  Evidently,  so  long  as  the  driv- 
ing engine  gives  constant  torque  and  the  machine  is  held  to  its 
synchronous  speed,  the  total  power  must  be  constant  irrespec- 
tive of  variation  of  the  excitation  of  the  alternator  field  magnets 
or  other  variables.  In  Fig.  410  the  triangle  OHB  is  similar  to 
the  triangle  OHB  in  Fig.  409.  In  this  case  the  power  given 
to  the  bus  bars  is  as  before 

P = El  cos  6. 

The  total  power  generated  which  is  to  be  kept  constant  is 
Pt  = EgI  cos  (/ 3 + 0 ), 

or  from  the  figure, 

Pt  = EJ  sin  ( 8 + 90°  - 0a). 

Therefore, 

Pt  = EgI  fsin  8 cos  (90°  — #a)+  cos  § sin  (90°  — 0O)}, 


SYNCHRONOUS  MACHINES 


683 


Fig.  410.  — Voltage  Locus  and  Diagram  of  a Generator  running  in  Parallel  when  its 

Excitation  is  varied. 


or 


Pt  = EgI( sin  8 sin  6a  + cos  8 cos  0a~). 
But  from  the  triangle  OVB  it  is  seen  that 
E sin  /3  = Ez  sin  8, 


or 


and 


or 


. c.  E sin  /3  E sin  8 
sm  o = — = t-, 


Ez  IZ„ 


Eg  = Ez  cos  8 + E cos  /3, 


s Ea  — E cos  /3 

cos  o - — 2 — • 

IZn 


The  symbols  used  here  are  the  same  as  in  the  discussion  of 
Fig.  409.  ’ Substituting  these  values  in  the  final  formula  given 
for  Pt,  there  results 


pt  = EJ KEPA  sin  0a  + EgIEg-EcoSl3 
±Zjn  1/Jn 


COS 


684 


ALTERNATING  CURRENTS 


or  P,  = E sin  /3  sin  6a  + cos  9a  — ^ E cos  /3  cos  0a, 

^ a ^ a a 

and  Pt  = - ^ E cos  (/3  + 9a)  + cos  0a. 

Z a A,. 


The  variables  in  this  expression  are  Eg  and  /3.  Multiplying 
Z 

through  by  - — to  give  a term  containing  only  Eg2,  there 
results 


PtZg  _ 

cos  6n 


E 

cos  6, 


Eg  cos  (yS  + 6a)  -f  E2. 


If  we  add  to  each  side  of  this 


E2 


4 cos2  6, 


, we  have 


PtZa  + E2 


cos  6n  4 cos2  6n  4 cos2  6n  cos  6, 


E - - E n Eg  COS  + 0a)  + E2, 


which  is  the  formula  for  a circle  expressed  in  polar  coordinates, 
where  the  left-hand  member  is  the  square  of  the  radius  and  where 

E 

the  initial  line  passes  through  the  center,  — — is  the  coordi- 

2 cos  6a 

nate  between  the  initial  point  and  the  center  of  the  circle,  Eg  is 
the  variable  coordinate,  and  (/3  + da)  is  the  angle  included  be- 
tween the  two.  Then,  to  construct  the  locus  of  Er  when  /3  is 
varied  by  changing  the  excitation  but  the  total  generated  elec- 
trical power  P,  remains  constant,  take  some  point  0 (Fig.  410) 
as  the  pole.  Lay  off  E = OV. \ vertically  for  convenience.  At 
the  angle  9a  to  the  right  of  0 Play  off  the  initial  line  XX'. 

E 

Measure  off  the  distance  — from  0 , giving  00' . With 

2 cos  0a 

O'  as  a center,  describe  the  arc  TT  having  a radius  equal  to  the 
square  root  of  the  sum  of  the  terms  on  either  side  of  the  polar 
equation.  This  is  the  required  locus  of  Eg , which  for  the  field 
excitation  that  gives  it  the  value  OB , shown  in  the  figure,  leads 
the  terminal  voltage  OP  by  the  angle  /3.  From  the  relation 

given,  it  is  seen  that  the  perpendicular  U O'  — — tan  6a , and 

E 

UO  = —;  therefore  O'  may  be  at  once  found  by  laying  off  the 


SYNCHRONOUS  MACHINES 


685 


E 

distance  — tan  6a  on  the  perpendicular  bisector  of  VO.  The  re-- 

quired  locus  is  then  found  as  before  by  striking  a circle  having 
the  radius 

PjZg  , ^ 

cos  0a  4 cos2  9a 


Since  the  total  power  generated  in  the  windings  is  kept  con- 
stant, the  output  will  be  a maximum  when  Ez(=  VB),  which 
varies  with  the  current  and  therefore  with  the  copper  loss,  is  at 
a minimum.  This  occurs  when  Ez  lies  on  a radius  of  the  locus 
TT  and  therefore  when  BV  extended  passes  through  O',  that 
is,  when  the  field  excitation  has  a value  which  makes  the  in- 
duced voltage  equal  to  OB'  and  the  drop  of  voltage  in  the 
armature  is  therefore  B'V.  From  the  geometrical  construction 
of  the  figure  it  can  also  be  seen  that  at  this  point  ER  takes  a 
position  on  OF"  extended  and  hence  that  the  external  power 
factor  is  unity.  If  the  excitation  be  made  less  than  when  Eg  = 
OB',  the  current  is  larger  and  leads  the  bus  bar  voltage  E\ 
and  when  the  excitation  is  increased,  as  when  Eg  = OB,  the 
current  lags  behind  E.  The  lagging  and  leading  components 
are  so  related  to  the  series  circuits,  comprised  of  the  generator 
under  consideration  and  the  other  generators  (which  maintain 
the  constant  terminal  voltage  E),  that  their  effects  tend  to  be 
respectively  those  of  leading  and  lagging  motoring  currents. 
Thus,  when  the  generator  voltage  is  low,  it  tends  to  cause  the 
voltage  of  other  machines  connected  to  the  line  to  fall  off 
and  when  high  to  increase.  This  is  explained  more  fully 
in  the  article  dealing  with  the  synchronous  motor.*  Substi- 

XT2 

tuting  w for  — H — — Eg  cos  (/3  + 0O)  + E 2 in  the  last 

4 cos2  6a  cos  0a 

equation  given,  an  expression  for  Pt  may  be  found  in  which  w 
stands  for  the  radius  of  the  circle  locus  of  Eg  with  fixed  power 
generated  (i.e.  circle  TT  of  Fig.  410,  which  may  be  called  a 

Power  circle),  thus  : 


Pt  = 


E2  \ 
4 cos2  6J 


Draw  a number  of  power  circles  concentric  with  TT  with  any 
convenient  radii,  such  as  T'  T' , T"  T",  and  T"'T'".  Then,  for 
any  given  or  fixed  voltage  Eg,  the  current  and  voltage  relations 


* Art.  167. 


686 


ALTERNATING  CURRENTS 


are  immediately  determinable  for  any  power.  Thus,  take  Et 
equal  to  OB , Fig.  410,  and  draw  a voltage  locus  MM  as  in  Fig. 
409 ; then  if  the  power  increases  from  that  corresponding  with 
radius  O' B to  that  corresponding  with  radius  O' B",  Bg  swings 
over  from  OB  to  OB"  and  the  current  swings  forward  from  the 
lagging  position  VN  to  the  leading  position  VN"  with  respect 
to  the  terminal  voltage. 

The  curves  of  current  for  constant  power  and  varying  exci- 
tation, when  plotted  with  generator  voltages  as  abscissas  and 
armature  currents  as  ordinates,  are  sometimes  called  V-curves 
from  their  shape.*  They  can  readily  be  obtained  from  the 
figure  by  giving  Eg  a series  of  values. 

Prob.  1.  Given  an  alternator  having  an  armature  resistance  of 
1 an  ohm  and  synchronous  reactance  of  4 ohms.  Construct  the 
locus  diagram  (Fig.  410)  of  the  generator  induced  voltage  when 
the  bus  bar  voltage  is  1000  volts  and  the  generated  power  is  100 
kilowatts.  When  the  power  is  50  kilowatts.  When  25  kilowatts. 

Prob.  2.  Determine  from  the  diagram  of  Prob.  1 the  power 
output  of  the  generator  when  the  total  powers  are  100,  50,  and 
25  kilowatts. 

163.  Maximum  Possible  Load  and  the  Regulation  of  Prime 
Movers  in  Parallel  Operation.  — An  engine  torque  greater  than 
the  resisting  torque  of  the  maximum  possible  electrical  load,  will, 
of  course,  cause  the  machine  to  break  from  synchronism.  With 
very  large  machines,  the  excessive  currents  that  result  from 
such  a condition  may  cause  injury  to  the  armature  conductors  by 
the  excessive  magnetic  forces  set  up  between  the  conductors  by 
the  large  currents  which  may  occur  before  the  circuit  breakers 
get  open ; for  at  the  ends  of  the  armature  core  where  the  wind- 
ings extend  beyond  the  core  the  conductors  are  sustained  in 
position  by  insulating  materials  of  comparatively  small  tensile 
strength.  This  is  similar  to  the  case  of  transformers,  f 

The  maximum  possible  load  which  a prime  mover  can  deliver 
to  its  alternator,  which  is  operating  in  parallel  with  others  under 
the  conditions  explained  in  the  last  article  where  dealing  with 
Fig.  410,  is  readily  determined  from  the  construction  of  that 
figure.  Thus,  with  fixed  field  excitation,  the  generated  voltage 
OB  swings  to  the  left  when  increased  power  is  supplied  from  the 


* Art.  170. 


t Art.  131. 


SYNCHRONOUS  MACHINES 


687 


prime  mover,  because  the  increased  driving  torque  tends  to 
make  the  machine  race.  The  vector  OB  therefore  cuts  larger 
and  larger  power  circles  ( TT \ T'  T',  etc.)  as  the  engine  power 
increases  until,  if  the  power  becomes  sufficiently  great,  it  lies 
upon  the  initial  line  or  radius  XX' . At  this  point  it  reaches 
the  largest  power  circle  possible  for  the  particular  field  excita- 
tion of  the  alternator.  Then  the  angle  /3  = 180°  — da.  Any 
greater  power  from  the  prime  mover  causes  the  point  B to  move 
toward  smaller  power  circles ; the  speed  increases  and  the  gen- 
erator falls  out  of  synchronism.  Reducing  the  excitation  may 
likewise  then  throw  the  generator  out  of  synchronism.  Thus, 
suppose  the  machine  is  running  under  constant  power  repre- 
sented by  the  power  circle  T"  T" . Then  suppose  that  the 
voltage  is  reduced  to  OB""  (Fig.  410),  making  the  triangle 
OVB ""  of  the  bus  bar  voltage  B = OV,  Bg  = OB" ",  and  the  in- 
ternal voltage  drop  Ez  — B""  V.  Any  smaller  excitation,  giving 
a smaller  value  of  OB"",  will  cause  the  point  B""  to  touch  a 
smaller  power  circle  and  the  engine  will  speed  up,  causing  the 
generator  to  break  from  synchronism.  The  excessive  currents 
caused  to  flow  in  either  of  these  limiting  cases,  as  indicated  by 
the  large  internal  drop  B""V  in  the  latter  and  an  even  larger 
drop  in  the  former  when  the  larger  constant  generated  voltage 
OB  was  used,  are  far  beyond  the  capacity  of  the  windings  of 
modern  commercial  alternators.  Further,  when  the  voltage 
drop  line  approximates  the  position  B""V,  the  reactive  effect 
upon  the  field  magnets  of  the  generators  connected  to  the  bus 
bars  would  be  very  great  and  cause  the  break  from  synchronism 
to  occur  before  the  voltage  vector  had  been  rotated  to  a point 
where  it  reached  the  line  XX'.  It  is  evident,  therefore,  that 
destructive  results  may  follow  an  excessive  lead  imposed  by  the 
prime  mover,  such  as  might  occur  through  accident  to  the  gov- 
ernor, or  great  weakening  of  the  field  current.  In  studying 
Figs.  408,  409,  and  410,  it  is  well  to  note  that  the  armature 
angle  of  lag  6a  has  been  made  smaller  than  is  likely  to  be  found 
in  commercial  machines. 

When  alternators  not  rigidly  connected  together  by  mechan- 
ical means  are  operated  in  parallel,  it  has  been  shown  above  that 
the  load  division  is  dependent  on  the  relative  torques  which  the 
prime  movers  tend  to  exert  upon  the  rotors  of  the  generators  to 
which  they  are  respectively  attached,  and  consequently  the 
division  of  load  between  alternators  operated  in  parallel  is  de- 


688 


ALTERNATING  CURRENTS 


pendent  on  the  speed  regulation  of  the  prime  movers.  If  one 
prime  mover  tends  to  lag  in  speed  behind  the  other,  it  produces 
less  power  than  is  the  case  if  it  tends  to  lead.  As  explained  * 
the  lagging  machine  under  these  circumstances  is  relieved  of  load 
by  the  leading  machine.  This  may  be  considered  to  be  due  to 
the  accelerating  effect  on  the  lagging  machine  caused  by  the  sj7n- 
chronizing  current,  or  to  the  resultant  reduction  of  load  on  the 
lagging  machine  and  its  consequent  increase  on  the  leading  ma- 
chine. The  instantaneous  torque  of  the  leading  machine  may  be 
considered  equal  to  that  caused  by  its  load  sent  to  the  external 
circuit  plus  the  torque  exerted  upon  the  lagging  machine  to  keep 
it  in  step,  while  the  torque  of  the  lagging  machine  (if  of  equal 
internal  impedance  and  equally  excited)  may  be  considered  equal 
to  that  of  an  equal  load  to  the  external  circuit  minus  the  torque 
expended  upon  it  by  the  leading  machine.  Evidently,  when  one 
of  two  equal  steam  engines  driving  alternators  in  parallel  has  its 
governor  set  for  a higher  speed  than  the  other,  it  acts  automati- 
cally and  takes  more  steam  when  held  below  its  normal  speed  by 
the  more  sluggish  second  engine,  while  the  second  engine,  being 
urged  forward  above  its  normal  speed  by  the  first  engine,  auto- 
matically takes  less  steam.  Thus,  the  torque  of  the  first  engine 
is  greater  than  that  of  the  second.  As  the  two  engines  are 
running  at  the  same  or  synchronous  speed,  the  first  must  then 
take  a larger  proportion  of  the  load  than  the  second. f It  is 
then  seen  that  the  division  of  load  on  alternators  in  parallel 
must  be  accomplished  by  regulating  the  governors  of  the  prime 
mover.  This  is  the  ordinary  practice.  The  governors  of  sev- 
eral engines  are  not  apt  to  regulate  exactly  alike,  and  the  par- 
allel operating  alternators  which  are  connected  to  them  are  apt 
to  divide  their  loads  unequally,  that  is,  one  machine  “ robs  ” the 
others,  unless  the  governors  are  adjusted  with  great  care.  It 
is  sometimes  desirable  to  fasten  the  governors  of  all  the  engines 
except  one  at  the  point  of  full  load  and  permit  the  governor  of 
the  one  engine  to  maintain  the  speed  regulation  for  all  and  carry 
the  variation  in  load.  Then  when  the  speed  tends  to  rise,  the 
governed  machine  tends  to  retard  the  others  by  absorbing  a 
cross  current,  that  is,  it  shifts  its  load  on  to  its  mates  and 
thereby  prevents  their  racing.  When  the  speed  tends  to  fall, 
the  governed  machine  tends  to  take  on  the  excess  load. 

* Art.  161.  t Art.  162. 


SYNCHRONOUS  MACHINES 


689 


164.  Effect  of  Form  of  Voltage  Curve  and  Variation  of  Angu- 
lar Velocity  on  Parallel  Operation  and  Division  of  Loads.  — The 

able  designer,  C.  E.  L.  Brown,  early  operated  his  own  alterna- 
tors with  smooth  iron  armature  cores  in  parallel  with  Ganz 
alternators  which  have  highly  reactive  pole-type  armatures.* 
The  former  machine  gave  a voltage  curve  which  approximated 
a sinusoid,  while  the  curve  given  by  the  latter  was  quite  irregu- 
lar and  peaked.  Dr.  Steinmetz  also  early  operated  machines 
with  smooth  iron  armature  cores  in  parallel  with  others  having 
toothed  cores,  f These  experiments  showed  that  machines  with 
different  voltage  curves  can  be  run  together  satisfactorily,  as  is 
now  very  frequently  done,  but  they  require  a considerably 


Fig.  411.  — Curves  Voltage  to  Cause  Flow  of  Harmonic  Cross  Currents  between 
Paralleled  Alternators  having  Voltage  Curves  like  A and  B. 

increased  exchange  of  current  as  compared  with  the  synchro- 
nizing current  of  machines  with  voltage  curves  which  are  exactly 
alike.  For,  since  the  unlike  curves  cannot  coincide  even  when 
the  machines  are  in  exact  step,  a current  of  more  or  less  irregu- 
lar form  must  be  exchanged  by  the  machines,  and  this  is  super- 
posed upon  the  true  synchronizing  current.  This  superposed 
current  may  have  a very  different  frequency  from  that  of  the 
machines  and  is  not  necessarily  wattless.  For  instance,  in  Fig. 
411  curves  A and  B represent  the  voltage  curves  of  two  alter- 
nators operating  in  parallel,  in  exact  step,  and  with  the  same 
effective  voltage.  The  instantaneous  differences  between  the 
two  curves  leave  residual  voltages  forming  the  curve  G which 
acts  to  set  up  a cross  current  in  the  series  circuit  comprising 
the  two  armatures. 

* Jour.  Inst.  E.  E .,  Vol.  22,  p.  600.  t Electrical  World,  Vol.  23,  p.  285. 


690 


ALTERNATING  CURRENTS 


Also,  higher  harmonic  cross  currents  which  may  flow  when 
polyphase  armatures  which  are  operating  in  parallel  act  much 
as  they  do  in  polyphase  induction  motors.*  The  third  harmonic 
sets  up  a rotary  armature  magneto-motive  force  opposed  to  that 
of  the  field  magnet.  It  has  already  been  pointed  out  that  the 
armature  currents  in  a polyphase  alternator  tend  to  set  up  a 
rotary  field  which  reacts  upon  the  alternator  field  magnet,  f 
When  the  cross  currents  have  sufficient  magnetizing  force,  they 
may  cause  the  machines  to  work  unsatisfactorily  in  parallel. 

The  angular  velocity  of  reciprocating  engines  used  for  driv- 
ing dynamos  is  by  no  means  uniform  throughout  the  revolu- 
tions, although  the  revolutions  may  be  isochronous.  Curves 
showing  the  instantaneous  crank  velocities  taken  from  engines 
of  well-known  types  are  often  quite  irregular.  If  alternators 
are  driven  from  separate  engines,  this  irregularity  of  angular 
velocity  may  be  the  cause  of  more  or  less  difficulty  in  parallel 
working,  and  some  machines  may  work  quite  satisfactorily  in 
parallel  when  driven  from  the  same  countershaft  or  from  sepa- 
rate hydraulic  or  steam  turbines  (which  give  a uniform  angu- 
lar velocity),  when  they  will  not  do  so  if  driven  by  separate 
reciprocating  engines.  It  is  evident  that  when  engines  of 
different  types  are  used  to  drive  alternators  that  it  is  desired 
to  parallel,  the  angular  velocities  may  vary  so  differently  for 
the  different  engines  that  the  synchronizing  current  may  be 
unable  to  hold  the  alternators  together.  It  is  therefore  desir- 
able to  specify,  in  obtaining  reciprocating  engines  for  parallel 
working,  a minimum  variation  in  angular  velocity.  This 
variation  should  not  cause  a displacement  of  the  rotor  from  its 
normal  position  of  more  than  two  electrical  degrees  during  the 
time  of  one  cycle.  $ 

Sometimes  it  is  necessary  to  cause  reciprocating  engines  to 
move  with  their  cranks  in  the  same  angular  space  relation, 
though  it  is  difficult  to  do  this,  and  it  results  in  the  variation 
in  frequency  due  to  variation  in  angular  speed  being  transmitted 
to  other  synchronous  apparatus  connected  to  the  system. 
These  other  machines  are  then  apt  to  hunt.§ 

165.  Synchronizers  and  Synchronizing.  — Mechanical  imper- 
fections in  engine  governors  and  machine  pulleys  cause  slight 

* Art.  198.  f Art.  152.  § Art.  173. 

J Standardization  Rules  of  Amer.  Inst,  of  Elect.  Eng.,  Rules  61-64. 


SYNCHRONOUS  MACHINES 


691 


differences  in  the  speeds  of  machines  intended  to  run  at  equal 
velocities.  Consequently  it  is  desirable  to  arrange  some  device 
for  determining  the  moment  a machine  is  in  synchronism  with 
one  with  which  it  is  to  be  thrown  in  parallel.  When  the 
alternators  are  to  be  connected  in  parallel,  the  terminals  of 
each  may  be  connected  directly  to  appropriate  bus  or  main 
conductors  through  convenient  indicating  instruments,  switches, 
and  safety  devices.  Before  switching  a new  machine  upon  the 
bus  conductors  it  should  be  brought  to  normal  speed  and  to 
the  voltage  of  the  other  machines.  Then  at  a moment  when  it 
is  in  synchronism  and  in  step  with  the  voltage  wave  of  the  bus 
bars,  it  may  be  switched  into  circuit  without  causing  a disturb- 
ance among  the  other  alternators. 


Fig.  412.  — Simple  Lamp  Synchronizer,  One  Phase  only  shown. 


Any  device  for  indicating  the  synchronous  relation  is  called 
a Synchronizer.  Its  simplest  form  for  low  voltage  machines 
consists  of  one  or  more  incandescent  lamps  in  series,  which  are 
connected  as  in  Fig.  412.  One  terminal  of  a phase  of  the 
alternator  is  connected  directly  to  a bus  conductor,  while  the 
other  is  connected  through  the  lamps  to  another  bus.  When 
the  voltage  waves  are  equal  but  not  in  opposition,  the  lamps  are 
illuminated,  and  at  the  moment  of  opposition  the  illumination 
dies  out.  If  the  frequencies  of  the  alternator  and  the  bus  bars 
differ  materially,  the  flashes  of  illumination  or  “ beats  ” are 
quite  rapid.  As  the  frequencies  approach  synchronism,  the 
beats  lengthen  out,  exactly  as  do  the  beats  of  two  tones  which 
are  approaching  unison.  The  alternator  should  be  connected  to 
the  bus  bars  at  an  instant  of  no  illumination  during  a period 


692 


ALTERNATING  CURRENTS 


when  the  beats  are  fairly  long.  This  indicates  that  the  voltage 
of  the  alternator  is  substantially  in  synchronism  with  that  of 
the  circuit  and  that  it  is  in  proper  step  or  angular  phase  posi- 
tion. Continued  darkness  of  the  lamps  under  these  circum- 
stances can  only  occur  when  the  machines  produce  the  same 
voltage  and  run  at  absolutely  the  same  frequencies,  which  is 
not  a practical  occurrence  unless  the  machines  are  rigidly 
connected  together. 

For  polyphase  machines  the  lamps  may  be  connected  to  a 
single  phase,  as  shown  in  Fig.  412,  since  if  one  phase  is  in  step 
with  the  bus  bars  the  others  must  be,  if  the  machines  are  prop- 
erly connected  to  the  bus  bar  leads.  When  connecting  a poly- 
phase machine  for  the  first  time  to  the  bus  bars,  much  care  must 
be  taken  to  give  assurance  that  the  various  phases  are  in  proper 
relation,  or  disastrous  results  may  follow  from  the  flow  of  ex- 
cessive currents  caused  by  the  reversed  relation  of  one  or  more 
phases  of  the  machine  with  reference  to  their  respective  bus 
bar  phases.  The  correct  connections  can  be  obtained  by  plac- 
ing  synchronizing  lamps,  or  equivalent  devices,  in  each  lead 
from  the  alternator  to  the  bus  bars.  When  the  connections 
have  been  properly  made,  the  synchronizers  will  show  opposition 
of  voltage  phase  between  each  alternator  lead  and  its  respective 
bus  bar.  Synchronizing  lamps  connected  as  in  Fig.  412,  placed 
in  each  lead,  should  all  be  dark  together. 

Since  alternators  are  commonly  built  for  high  voltages,  it  is 
usual  to  use  a transformer  with  the  synchronizer  lamps.  The 
primary  winding  of  this  transformer  may  be  composed  of  either 
one  or  two  circuits.  When  it  is  composed  of  one  winding,  one 
terminal  of  the  alternator  is  connected  to  a bus  bar  through  it, 
the  other  terminal  of  the  alternator  being  connected  directly 
to  the  other  bus  bar  (Fig.  413).  In  this  case  the  lamp  on 
the  secondary  circuit  acts  exactly  as  the  synchronizer  lamps 
already  described.  When  the  primary  winding  is  composed  of 
two  circuits,  one  is  connected  between  the  bus  conductors  and 
the  other  between  the  terminals  of  the  alternator  (Fig.  414). 
In  this  case  the  proper  phase  relation  for  switching  the  alternator 
into  circuit  may  be  indicated  either  by  darkness  or  by  illumina- 
tion of  the  lamps,  depending  upon  the  connections  of  the  pri- 
mary windings  of  the  synchronizer  transformer.  This  arrange- 
ment is  advantageous,  since  it  allows  the  use  of  a double-pole 


SYNCHRONOUS  MACHINES 


693 


switch  at  the  dynamo,  while  the  previously  described  arrange- 
ments require  the  use  of  single-pole  switches.  The  latter  ar- 
rangement may  be  modified  by  using  two  separate  transformers, 


Fig.  413.  — Synchronizer  Lamp  connected  to  the  Secondary  Winding  of  a Simple 

Voltage  Transformer. 

the  primary  circuit  of  one  being  connected  to  the  alternator 
and  that  of  the  other  to  the  bus  bar  circuit.  The  secondary 
windings  are  connected  together  in  series  with  one  or  two  lamps. 


L 

Fig.  414.  — Sychronizer  Lamp  connected  to  the  Secondary  Winding  of  a Transformer 
having  Two  Primary  Coils. 


694 


ALTERNATING  CURRENTS 


If  the  secondary  windings  are  connected  directly  in  series,  as  in 
Fig.  415,  darkness  of  the  lamps  indicates  the  instant  for  con- 
i necting  the  alternator 


to  the  bus  conductors. 
_ If  the  secondaries  are 
cross-connected  as  in 
Fig.  416,  maximum 
illumination  of  the 
lamps  indicates  the 
moment  when  the  ma- 
chines are  in  proper 
step.  In  this  country 
the  general  practice 
has  been  to  connect 
lamp  synchronizers  so 
that  darkness  indicates 
when  the  machines  are 

Fig.  415.  — Synchronizer  Lamps  connected  in  the  Sec-  in  step.  1 his  has  the 
ondary  Circuit  of  Two  Transformers  having  their  evident  advantage 
Secondary  Windings  in  Series. 

that  darkness  is  a con- 
dition which  is  more  readily  distinguished  than  the  condition  of 
maximum  brightness.  This  practice,  however,  is  not  always 


Ai 


bvVvWi/vV\KWvV/vV 

-A/WV VvV-| 

— o — o 


followed. 

In  any  of  these 
methods,  the 
lamps  may  be 
replaced  by  a 
sensitive  high 
resistance  alter- 
nating-current 


amperemeter  or 
galvanometer 
which  is  dead- 
beat. 

An  ingenious 
device  for  use  as 
a synchronizer, 
which  consists  of 
two  electromag- 
nets made  with 


L^WWVWWWA 


— o — o 

Fig.  416. — Synchronizer  Lamps  arranged  with  Two  Trans- 
formers, where  One  of  the  Secondary  Windings  is  reversed 
with  Respect  to  the  Other. 


SYNCHRONOUS  MACHINES 


695 


iron  wire  cores,  has  been  used  sometimes.  The  windings  oi 
these  are  connected  respectively  to  the  alternator  and  the  bus 
bar  circuit.  Each  magnet  has  placed  in  front  of  it  an  iron 
diaphragm  which  emits  a tone  which  has  a pitch  due  to  the 
frequency  of  the  current  flowing  in  the  winding.  In  front 
of  the  magnets  are  placed  resonators  which  magnify  the  sound 
emitted  by  the  diaphragms.  When  the  two  tones  are  not  in 
exact  harmony,  interference  causes  beats,  and  the  synchronizer 
emits  an  intermittent  sound.  If  the  speeds  of  the  machines  are 
brought  nearer  and  nearer  to  synchronism,  the  beats  become 
less  rapid.  At  exact  synchronism,  the  beats  die  out  and  a 
clear  tone  results.  The  alternator  may  or  may  not  be  in  step 
when  the  clear  tone  is  given,  but  a small  machine  may  be 
safely  thrown  into  circuit,  for  the  interaction  of  the  current 
waves  will  bring  it  into  proper  phase  relations.  This  synchro- 
nizer was  designed  for  use  with  synchronous  motors.  It  cannot 
give  satisfaction  with  large  alternators  or  with  alternators  that 
are  furnishing  current  at  constant  voltage  for  incandescent 
lighting,  since  placing  an  alternator  in  the  circuit  while  it  is 
out  of  step  is  likely  to  momentarily  disturb  the  voltage.  A 
visual  synchronizer,  having  much  the  same  principle,  consisting 
of  a set  of  vibrating  reeds,  can  also  be  constructed. 

It  is  possible  to  do  without  a synchronizer  when  connecting 
small  alternators  in  parallel,  provided  they  are  driven  at  ap- 
proximately equal  speeds.  In  this  case  if  the  alternator  is 
brought  to  the  proper  voltage  and  is  then  connected  to  the  bus 
bars,  the  machine  reactions  will  bring  it  into  step.  This  plan 
was  practiced  to  some  extent  in  one  or  two  earlier  American 
plants,  but  it  is  vicious  in  its  working  unless  inductance  coils 
of  considerable  magnitude  are  introduced  in  circuit  with  the 
incoming  alternator.  Throwing  machines  on  to  the  bus  bars 
without  synchronizing  usually  causes  a disturbance  in  the  vol- 
tage, and  it  also  strains  the  armature  even  of  a small  alternator 
on  account  of  the  sudden  torque  impressed  upon  it  to  bring  it 
into  step,  and  in  large  machines  is  apt  to  cause  destructive  strains 
in  the  ends  of  the  armature  coils.  In  order  to  prevent  an  undue 
rush  of  current  when  alternators  are  thrown  together  while 
slightly  out  of  step,  external  inductance  coils  are  sometimes 
inserted  temporarily  in  series  between  the  machines  and  the 
bus  bars. 


696 


ALTERNATING  CURRENTS 


A transformer  having  three  legs  and  windings  like  the  tri- 
phase transformer  of  Fig.  294  may  be  used  for  operating  the 
synchronizing  indicator.  The  windings  of  the  two  outer  legs 
are  attached  respectively  to  the  bus  bars  and  the  machine,  and 
a voltmeter  or  lamp  is  connected  across  the  middle  winding. 
At  the  point  of  synchronism  the  magnetic  fluxes  produced  by 
the  two  exciting  coils  are  in  parallel  in  the  middle  core  and  the 
voltmeter  shows  a maximum  reading. 

An  instrument  which  indicates  whether  the  machine  to  be 
synchronized  is  running  fast  or  slow,  as  well  as  the  condition 
of  step,  is  sometimes  called  a Synchroscope  or  Phase  indicator. 
One  of  the  simplest  arrangements  of  this  kind  consists  of  a disk 
attached  to  the  alternator  shaft  or  that  of  a small  synchronously 
running  motor.  This  disk  is  painted  in  alternately  white  and 
black  sectors,  each  sector  having  an  angular  width  equal  to  the 
polar  pitch.  In  front  of  the  disk  is  placed  an  arc  lamp  at- 
tached to  the  bus  bars  through  the  necessary  transforming 
device.  The  ai'rangement  is  the  same  as  that  sometimes  used 
for  measuring  tire  slip  in  an  induction  motor.*  Then  when 
the  illuminated  sectors  of  the  disk  apparently  stand  still,  the 
frequency  of  the  machine  is  the  same  as  that  of  the  bus  bars. 
When  the  illuminated  sectors  of  the  disk  apparently  move  in 
the  direction  opposite  to  the  direction  of  the  rotation  of  the 
rotating  part  of  the  machine,  the  speed  of  the  latter  is  too  slow  ; 
and  when  the  illuminated  sectors  apparently  move  in  the  direc- 
tion of  the  rotating  part,  the  speed  of  the  machine  is  too  fast. 
The  step  phase  when  the  bus  bars  and  machine  are  in  synchro- 
nism is  indicated  by  the  position  of  the  stationary  sectors,  but 
the  additional  use  of  a lamp  synchronizer  is  desirable. 

In  a three-phase  machine  the  use  of  three  lamps,  as  shown  in 
Fig.  417,  may  be  adopted.  In  this  case  the  three  lamps,  through 
suitable  transforming  devices,  are  placed  at  the  corners  of  a tri- 
angle and  a terminal  of  each  connected  to  one  of  the  three  bus 
bars,  while  the  other  three  terminals  are  connected  to  one  phase 
wire  of  the  machine.  When  the  machine  is  in  synchronism, 
the  lower  right-hand  lamp  in  the  figure  is  in  darkness  while 
the  other  two  are  equally  bright.  When  the  speed  of  the  ma- 
chine is  slightly  too  high,  the  lamps  glow  one  after  the  other, 
giving  the  appearance  of  a rotation  in  one  direction  ; and  when 


* Art.  189. 


SYNCHRONOUS  MACHINES 


697 


BUS  BARS 

17 

SWITCH  / 

\ 

V 

l 

LAMPS 

Fig.  417. 


ALTERNATOR 


A Three-lamp  Synchroscope  for  a 
Three-phase  System . 


the  speed  is  too  low,  the  rotation  of  the  lighting  is  in  the  other 
direction.  Various  modified  arrangements  to  accomplish  this 
result  may  be  made. 

Synchroscopes  using  the  magnetic  relations  of  a movable  coil 
to  which  a pointer  is 
attached  are  now  gener- 
ally used  in  commercial 
plants  and  are  eminently 
convenient  and  satisfac- 
tory. A common  form 
has  an  armature  which 
is  somewhat  similar  to 
the  field  magnet  of  a two- 
coil,  split-phase  induction 
motor  without  its  iron 
core.  A two-pole  field 
magnet,  0,  embracing 
the  armature  may  have 
its  coil  connected  to  the 
bus  bars  (Fig.  418),  and 
the  split-phase  armature  winding,  its  two  coils  A and  B wound 
at  right  angles  and  its  shaft  pivoted  between  the  pole  pieces 

of  the  field,  may  be  connected 
to  the  machine  to  be  synchro- 
nized, or  vice  versa.  A pointer 
is  attached  to  the  armature  for 
the  purpose  of  indicating  its 
motion.  When  the  currents  in 
the  field  winding  and  armature 
windings  are  in  synchronism, 
the  armature  takes  an  angular 
position  with  reference  to  the 
field  magnet  which  is  deter- 
mined by  the  phase  difference 
between  the  currents  entering 
the  field  and  armature  wind- 
ings, the  zero  position  being 
assumed  when  the  currents  are 
in  step  with  each  other.  When  the  frequency  of  the  machine 
is  greater  than  that  of  the  bus  bars,  the  armature  of  the  syn- 


LEADS  TO 
ALTERNATOR 

wmrwrci 


Fig.  418.- 


U&flOOQPOQ  J 
^LEADTa 

Mil  {goo 
BUS  BARS 

-Diagram  of  a Synchroscope  of 
Split-phase  Type. 


698 


ALTERNATING  CURRENTS 


chronizer  rotates  in  one  direction ; and  when  the  frequency  of 
the  bus  bar  voltage  is  the  greater,  the  rotation  is  in  the  opposite 
direction.  Thus,  when  the  machine  is  in  exact  step  and  syn- 
chronism with  the  bus  bar  voltage,  coil  A takes  a position  with 
its  phase  at  right  angles  to  the  flux  of  the  field.  B is  then 
inert,  its  time  phase  being  90°  from  that  of  the  field.  When 
the  phases  change,  say  45°,  A and  B then  equally  tend  to  place 
themselves  at  right  angles  to  the  field  flux,  and  the  armature 
must  take  up  a position  with  A 45°  from  its  former  position. 
At  90°  difference  between  the  machine  and  bus  bar  phases,  A is 
inert  and  B takes  a position  at  right  angles  to  the  field  flux. 
At  180°  difference  of  phase  A takes  a position  180°  from  its  first 
position,  and  so  on.  Then  if  the  machine  voltage  is  in  synchro- 
nism but  out  of  step  with  the  bus  bar  voltage,  the  pointer  takes 
a definite  angular  position ; but  if  the  machine  frequency  is 
different  from  that  of  the  bus  bars,  the  pointer  must  make  one 
complete  revolution  in  one  direction  for  each  cycle  gained  and 
in  the  other  direction  for  each  cycle  lost. 

Considering  the  two  coils  A and  B wound  on  the  armature 
at  right  angles  to  each  other  and  traversed  by  currents  in  quad- 
rature, then  the  effecf-  of  the  windings  is  to  set  up  a rotating 
magnetic  field  which  makes  its  complete  rotation  with  a fre- 
quency equal  to  the  frequency  /'  of  the  currents  in  the  coils.* 
The  field  magnet  directs  a flux  through  the  armature  which 


has  the  frequency  / of  the  current  through  the  field  winding. 
The  armature  tends  to  place  itself  in  such  a position  that  its 
flux  shall  be  at  all  times  in  the  same  direction  as  the  flux  of  the 


field  magnet,  but  during  one  period  of  the  field  current  the  ro- 

/' 

tating  field  of  the  armature  will  rotate  through  ‘y  860°,  where 


/is  the  frequency  of  the  current  in  the  field  coil  and/'  the  fre- 
quency of  the  currents  in  the  armature  coils.  Therefore  the 

armature  core  will  turn  through  an  angle  of  fl — ^ \ 360°  = 


f-f 

f 


360°  during  each  period  of  the  field  flux,  which  is  equiv- 


alent to  saying  that  the  armature  will  rotate  /— /'  revolutions 
per  second  since  the  field  flux  makes/ complete  cycles  per  sec- 
ond. If  /'  is  smaller  than  /,  the  armature  core  will  rotate  in 
the  same  direction  as  the  rotating  magnetic  field  set  up  by  its 


* Art.  186. 


SYNCHRONOUS  MACHINES 


699 


windings,  but  at  a speed  which  is  slower  in  the  ratio  of 


/-/' 

f 


and  if  f is  the  larger,  the  armature  will  rotate  in  the  opposite 

f-f 


direction,  but  also  with  the  speed  ratio 


f 


The  bus  bars 


being  permanently  connected  to  one  of  the  synchroscope  cir- 
cuits, the  machine  to  be  synchronized  must  be  connected  to  the 
other  circuit,  and  the  direction  of  rotation  of  the  needle  then 
always  indicates  whether  the  incoming  machine  is  running  too 
fast  or  too  slow.  When  the  pointer  of  the  instrument  stands 
still,  the  two  frequencies  are  equal.  If  the  currents  in  A and 
B stand  respectively  in  phase  with  and  90°  behind  the  voltage 
impressed  on  their  joint  circuit  and  the  field  flux  is  45°  behind 
the  voltage  impressed  on  the  field  magnet  circuit,  the  pointer 
will  stand  at  zero  when  the  voltages  of  bus  bars  and  machine 
are  in  both  synchronism  and  step.  The  two  differing  phases 
can  be  obtained  for  the  armature  coils  by  the  use  of  inductance 
and  resistance  as  shown  in  the  figure,  or  where  the  system  is 
polyphase,  directly  from  the  machine  or  bus  bars. 

Various  other  arrangements  for  accomplishing  the  result  de- 
scribed in  the  preceding  paragraph  are  also  possible. 

Automatic  synchronizers  in  which  the  main  switch  is  closed 
by  the  action  of  an  electro-magnet  with  its  coil  in  the  synchro- 
nizing circuit  are  sometimes  used,  though  they  have  not  fully 
shown  their  desirability.  The  arrangement  usually  comprises 
an  electro-magnet  in  series  with  the  synchronizer  lamp  circuit, 
the  armature  of  which  either  directly  trips  the  main  switch, 
which  closes  by  a spring,  or  closes  a switch  which  in  turn 
causes  a strong  electro-magnet  attached  to  the  main  switch  to 
become  excited,  and  through  the  attraction  of  it  the  armature 
closes  the  main  switch. 

166.  Methods  of  Connecting  Alternators  in  Parallel  to  Feeder 
Circuits.  — Switchboards  for  connecting  alternators  in  parallel 
are  almost  as  various  in  details  of  construction  as  are  the  num- 
ber of  switchboards  built,  that  is,  each  board  is  designed  to 
meet  the  conditions  it  must  fill,  although  it  is  usual  to  follow 
general  standards.  They  are  usually  of  much  the  same 
character  so  far  as  relates  to  the  underlying  duty  they  are  to 
perform.  Figure  419  is  the  diagram  of  a suitable  board  designed 
by  one  of  the  large  manufacturing  companies  for  controlling 


700  ALTERNATING  CURRENTS 


Fig.  41!).  — Switchboard  Connections  for  operating  Three  Tri-phase  Generators  in  Parallel. 


SYNCHRONOUS  MACHINES 


701 


Fig.  420.  — Cross  Section  of  an  Alternating-cur- 
rent Switchboard. 


tri-phase  alternators.  The 
lower  heavy  lines  repre- 
sent the  exciter  bus  bars 
and  connections,  the  upper 
heavy  lines  represent  the 
alternating  current  bus 
bars  (which  are  often  du- 
plicated for  safety  but  are 
shown  here  in  only  one 
set)  and  connections,  and 
the  light  lines  show  the 
measuring  instrument  con- 


nections.  The  feeders  to 
the  load  are  shown  at  the 
right,  and  the  synchroniz- 
ing apparatus  and  volt- 
meters to  the  left.  The  details  of  the  connections  may  be  readily 
obtained  from  a study  of  the  figure.  Additional  generators  and 
feeders  can  be  added  to  the  system  shown  in  the  figure  by  ex- 
tending the  bus  bars.  Ordinarily  all 
high  voltage  is  kept  away  from  the  face 
of  the  switchboard,  and  the  highest 
voltage  switches  are  usually  situated 
at  a distance  in  fireproof  cells  and  oper- 
ated by  suitable  mechanical  or  electri- 
cal transmission,  the  high  voltage  bus 
bars  being  located  in  fireproof  ducts. 
Thus  Fig.  420  shows  a cross  section  of 
a switchboard  in  which  the  high  vol- 
tage oil  switches  and  the  current  and 
voltage  transformers  for  operating  the 
switchboard  instruments  are  placed  on 
a framework  at  the  rear  of  the  board. 
The  instruments  and  controlling 
levers  are  in  front  of  the  control 
board.  The  switches  may  be 
made  to  act  as  automatic  circuit 
breakers  by  adding  suitable  auto- 


Fig.  421.  — View  of  High  Voltage  Auto- 
matic Switch. 


matic  tripping  devices.  A 
switch  or  circuit  breaker  with 


702 


ALTERNATING  CURRENTS 


its  parts  immersed  in  oil,  called  an  oil  switch  or  circuit  breaker, 
suitable  for  voltages  of  several  tens  of  thousands  of  volts  is 
shown  in  Fig.  421,  and  in  Fig.  422  is  a cross  section  of  the 
same  apparatus.  The  contact  parts  are  immersed  in  oil  in  the 

large  tank  and  the  con- 
tact is  opened  and  closed 
through  the  intervention 
of  the  wooden  connect- 
ing rod  A.  Movement 
for  operating  the  switch 
is  transmitted  to  the  rod 
A by  means  of  the  sole- 
noid B and  associated 
links.  The  current  for 
the  solenoid  is  controlled 
from  the  control  board. 
A small  electric  motor 
may  be  substituted  for 
the  solenoid.  The  ter- 
minals for  leading  current  into 
and  out  of  the  switch  are  shown 
at  TT , Fig.  421. 

167.  Alternators  as  Synchro- 
nous Motors.  — Any  alternator 
may  be  run  as  a motor,  provided 
it  is  brought  up  to  synchronous 
speed  and  into  step  before  it  is 
thrown  into  circuit.  The  motor 
will  then  run  in  complete  syn- 
chronism if  left  to  itself.  If  it 
is  overloaded,  or  by  other  means 
is  thrown  out  of  synchronism,  it 
will  stop.  In  general,  the  action 
Fig.  422. — Section  of  High  Voltage  of  an  alternator  used  as  a S}-n- 
Switch-  chronous  motor  is  quite  similar 

to  that  of  an  alternator  operated  in  parallel  with  another. 
A great  disadvantage  of  single-phase  synchronous  motors 
is  the  fact  that  they  are  not  self-starting : but  must  be 
brought  up  to  speed  before  they  will  operate.  The  starting  of 
a single-phaser  may  be  done  by  a small  series-wound  auxiliary 


SYNCHRONOUS  MACHINES 


703 


motor  made  with  laminated  fields.*  A small  two-pliase  motor 
with  a device  for  splitting  the  current  into  two  phases  may 
be  used,  or  the  exciter  of  the  alternator  may  be  run  as  a motor 
from  a direct  current  source  and  used  to  bring  the  alternator 
into  synchronism. 

A polyphase  synchronous  motor  may  be  started  by  an  ordinary 
polyphase  induction  motor  coupled  to  its  shaft ; or  it  may  be 
self-started  as  an  induction  motor.  In  the  latter  case  the  ar- 
mature of  the  synchronous  machine  becomes  the  primary  portion, 
and  the  field  magnet  the  secondary  portion,  of  the  rotating  field 
induction  motor.  For  starting,  the  field  circuit  is  usually 
opened.  Current  is  permitted  to  flow  from  the  main  circuit 
at  reduced  voltage  through  the  armature  windings,  an  auto- 
transformer (Fig.  423)  or  a transformer  having  several  voltage 
taps  from  the  secondary  winding  being  ordinarily  used  for  the 
purpose  of  giving  the  reduced  voltage.  When  the  machine  has 
been  brought  up  to  speed  by  the  torque  exerted  between  the 
eddy  currents  induced  in  the  pole  faces  and  the  rotating  magnetic 
field  set  up  by  the  currents  in  the  armature  windings,  the  field 
switch  may  then  be  thrown  on  to  the  direct  current  exciter  bus 
bars,  and  the  machine  will  continue  to  rotate  as  a synchronous 
motor.  When  the  field  coils  are  large,  the  winding  must  be 
opened  in  a number  of  places  to  prevent  excessive  voltages 
being  generated  in  them  by  the  rotating  armature  flux.  When 
the  rotor  of  the  machine  has  been  brought  to  as  high  a speed  as 
possible  as  an  induction  motor,  it  may  jump  into  synchronism 
before  the  field  switch  is  closed  to  send  direct  current  through 
the  field  windings.  The  action  is  similar  to  that  in  large  un- 
loaded induction  motors  f with  appreciable  polar  projections, 
that  is,  the  magnetic  flux  from  the  armature  rotating  but  very 
slightly  faster  than  the  projecting  poles  of  the  field  magnet, 
magnetizes  the  poles  as  it  passes  over  them  and  they  take  up 
synchronous  speed,  continuing  to  obtain  magnetization  from  the 
armature  magneto-motive  force.  When  the  field  switch  is  closed, 
the  magnetic  flux  already  set  up  in  the  fields  by  the  armature 
reaction  may  be  in  the  wrong  direction,  in  which  case  the  arma- 
ture must  fall  back  the  distance  of  the  polar  pitch  before  the 
regular  field  flux  can  produce  forward  operating  torque.  The 
processes  for  starting  rotary  converters  are  given  in  greater 
* Chap.  XII.  + Ibid. 


704 


ALTERNATING  CURRENTS 


detail  later,*  and  as  they  are  essentially  the  same  on  the  alter- 
nating current  side  as  for  synchronous  motors,  many  details  con- 
cerning starting  the  latter  are  deferred  for  the  present. 

Figure  423  shows  the  elementary  connections  for  starting  a 
quarter-phase  synchronous  motor  when  an  autotransformer  con- 
trolled by  a cylinder  switch  is  used.  The  rheostat  in  the  motor 
field  circuit,  the  main  switch,  and  measuring  instruments  are 
not  shown.  The  exciter  may  be  driven  mechanically  by  the 
synchronous  motor  and  should  build  up  its  voltage  while  the 


Fig.  423.  — Elementary  Diagram  of  Connection  for  a Quarter-phase  Synchronous 

Motor. 


synchronous  motor  is  coming  up  to  speed.  In  order  to  make 
the  machine  field  frame  assume  the  function  of  the  secondary 
conductors  of  a short-circuited  induction  motor,  it  is  sometimes 
desirable  to  place  copper  conductors  about  the  field  poles  in 
such  a manner  as  to  make  short-circuited  paths  (Fig.  442). 
When  the  motors  are  very  large,  or  are  not  designed  to  start  as 
induction  motors  without  an  excessive  flow  of  current,  they  are 
commonly  brought  into  synchronism  by  starting  motors, f as  are 
single-phase  machines. 

168.  Relation  of  Voltages,  Currents,  and  Power  in  Synchronous 
Motors.  Synchronous  Impedance.  — When  a synchronous  motor 
is  put  in  the  circuit,  a peculiar  relation  exists  between  the 
strength  of  the  field  of  the  motor  and  the  current  in  its  arma- 
ture. In  direct-current  motors,  if  the  strength  of  the  field  is 


* Art.  181. 


f Chap.  XII. 


SYNCHRONOUS  MACHINES 


705 


slightly  changed  without  altering  any  of  the  other  conditions, 
the  speed  of  the  motor  changes  inversely,  and  the  current  in 
the  armature  remains  practically  unchanged ; but  the  speed  of 
a synchronous  motor  cannot  change  permanently,  and,  con- 
sequently, upon  first  consideration,  it  would  appear  that  the 
strength  of  the  field  magnet  of  a synchronous  motor  must  be 
exactly  adjusted,  in  order  that  the  machine  may  operate  satis- 
factorily. This  is  not  the  case,  however,  as  was  seen  in  the 
discussion  of  the  parallel  operation  of  alternators,*  on  account 
of  the  effect  which  may  be  gained  through  variations  of  the 
relative  phases  of  the  current  and  of  the  impressed  and  counter 
voltages.  The  active  voltage,  which  at  any  instant  causes 
current  to  flow  through  the  armature  of  a motor,  is  equal  to  the 
difference  of  the  corresponding  instantaneous  values  of  the  im- 
pressed and  counter  voltages.  If  the  field  strength  of  a motor 
is  so  adjusted  that  the  value  of  the  impressed  and  counter  vol- 
tages are  equal,  and  the  motor  armature  is  brought  by  external 
means  into  exact  step  with  the  impressed  voltage  curve,  then, 
when  the  motor  is  switched  on  the  supply  main,  it  will  fall 
back  in  phase  with  respect  to  the  impressed  voltage  sufficiently 
to  permit  the  proper  driving  current  to  pass  through  the  arma- 
ture. Now  suppose  that  at  some  instant  the  load  is  increased, 
the  difference  of  instantaneous  voltage  at  that  instant  will  be 
insufficient  to  cause  the  larger  current  to  flow  through  the 
armature  which  is  necessary  to  produce  the  torque  for  the  new 
load.  The  motor,  therefore,  falls  back  in  its  phase  or  armature 
step  without  losing  synchronism,  if  the  load  is  not  too  great, 
and  then  continues  operating  in  synchronism,  but  with  a greater 
lag  in  step.  When  a motor  lags  in  step  behind  the  phase  of 
impressed  voltage,  its  counter-voltage  lags  to  an  equal  extent. 
The  armature  current  ordinarily  takes  an  intermediate  phase, 
so  that  it  is  behind  the  resultant  voltage  but  in  advance  of  op- 
position to  the  counter-voltage. 

Were  it  not  for  the  just  described  automatic  adjustment  of 
the  phases  of  the  impressed  and  counter-voltages,  it  would  be 
necessary  to  adjust  the  field  excitation  of  a synchronous  motor 
so  that  its  counter-voltage  would  be  less  than  the  impressed 
voltage  and  the  range  of  load  carried  with  a given  excitation 
would  be  limited  to  one  value.  The  effects  due  to  the  auto- 

* Arts.  162,  163. 

2 z 


706 


ALTERNATING  CURRENTS 


matic  adjustment  of  phase,  however,  make  it  possible  to  fix  the 
field  excitation  once  for  all,  and  yet  have  the  motor  operate 
on  widely  varying  loads.  It  is  even  possible,  on  account  of 
the  automatic  adjustment  of  the  voltage  phases,  to  operate  a 
motor  when  its  excitation  is  much  greater  or  much  less  than  its 
normal  value. 

The  adjustment  is  assisted  by  the  effect  of  armature  reactions 
on  the  motor,  in  which  a current  lagging  with  respect  to  the 
impressed  voltage  tends  to  strengthen  the  field  magnet  and  a 
current  leading  with  respect  to  the  impressed  voltage  tends  to 
weaken  the  magnet.*  When  a single  motor  is  operated  from 
an  alternator  of  about  its  own  size,  the  automatic  adjustment 
of  the  machines  is  still  more  marked,  since  the  current  which 
strengthens  the  field  magnet  of  the  motor  tends  to  weaken  that 
of  the  alternator  as  the  load  is  varied,  and  vice  versa.  A balance 
for  each  load  is  reached  when  the  two  voltages  have  attained 
such  values  that  their  vector  difference  or  resultant  is  just  suf- 
ficient to  drive  a current  through  the  circuit  which  gives  a 
torque  sufficient  to  keep  the  motor  running  synchronously. 

It  is  evident  from  the  preceding  paragraphs  that  the  armature 
current  of  a synchronous  motor  must  have  a quadrature  com- 
ponent which  depends  upon  the  phase  difference  of  the  im- 
pressed and  counter  voltages  and  the  internal  angle  of  lag,  and 
it  may  readily  be  seen  that  a particular  excitation  of  a syn- 
chronous motor  field  magnet  reduces  the  armature  current  to 
a minimum  (or  makes  the  power  factor  a maximum)  when  the 
motor  is  carrying  a particular  load. 

The  current  which  flows  through  the  armature  is  not  only  de- 
pendent upon  the  ordinary  impedance,  consisting  of  the  resist- 
ance of  the  windings  and  their  self-inductances,  but  also  upon 
the  field  reactions  caused  by  the  quadrature  component  of  cur- 
rent. For  convenience  the  effect  of  the  field  reactions  is  fre- 
quently included  and  treated  as  part  of  the  armature  impedance 
instead  of  as  a factor  directly  affecting  the  value  of  the  counter- 
voltage  of  the  motor.  The  impedance  then  obtained  is  called 
synchronous  impedance.  As  explained  earlier  f it  may  be  de- 
fined as  the  ratio  of  the  voltage  which  would  be  developed  at 
the  given  excitation,  if  the  machine  was  run  as  a generator  at 
synchronous  speed  without  current  in  the  armature,  to  the  am- 

fArt.  156. 


* Art.  152. 


SYNCHRONOUS  MACHINES 


707 


peres  that  would  flow  though  the  armature  at  the  same  excita- 
tion when  it  is  short-circuited.  In  the  diagrams  for  the  parallel 
operation  of  alternators  and  those  to  follow  for  the  synchronous 
motor  the  machine  voltages  used  are  those  that  would  be  gen- 
erated were  field  reactions  absent,  while  to  compensate  for  this 
assumption,  the  synchronous  impedance  of  the  armature,  which 
includes  the  effect  of  reactions,  has  been  used  instead  of  the  or- 
dinary impedance  caused  only  by  the  self -inductance  and  resist- 
ance of  the  armature  windings.  The  synchronous  impedance 
has  also,  for  simplicity,  been  considered  constant  when  the  ex- 
citation has  been  varied.  As  a matter  of  fact  the  field  reactions 
and  the  true  reactance  of  the  armature  vary  with  the  space  posi- 
tion of  the  armature  coils  with  reference  to  the  field  magnet  and 
the  time  phase  of  the  current,  as  well  as  upon  the  reluctance  of 
the  leakage  magnetic  circuits,  so  that  some  error  is  involved  in 
assuming  the  synchronous  impedance  to  be  constant.  Thus,  it 
is  evident  that  the  synchronous  impedance  is  less  for  very  high 
excitations  than  for  those  of  low  values,  since  in  the  first  case  the 
magnetic  circuits  are  more  highly  saturated  and  a given  arma- 
ture magneto-motive  force  has  less  effect  upon  the  field  mag- 
netic flux  than  in  the  second  case. 

In  order  to  bring  out  more  clearly  the  relation  of  voltages 
and  currents  in  the  motor,  recourse  may  be  had  to  a diagram 
in  which  relations  of  current  and  voltage  are  shown  much  as 
in  parallel  working.  It  was  shown*  that  in  parallel  working 
the  machines  were  held  in  step  by  a motor  action,  caused  by 
a wattless  current  flowing,  and  that  if  one  machine  was  cut 
off  from  its  prime  mover  it  would  continue  to  run  in  synchro- 
nism as  a motor.  In  Fig.  424  let  01  be  the  current  flowing 
from  a circuit  through  the  armature  of  an  alternator  acting  as 
a motor,  let  SR  be  the  drop  of  voltage  in  the  synchronous 
reactance,  equal  and  opposite  to  the  reactive  voltage  OL  of  the 
armature,  and  let  OS  be  the  drop  of  voltage  in  the  armature 
resistance ; then  OR  is  the  resultant  voltage  IZ , required  to 
act  on  the  synchronous  impedance  of  the  armature  to  pass  the 
current  01  through  the  circuit.  Suppose  the  voltage  OEx  is 
impressed  at  the  armature  terminals,  and  the  motor  is  excited 
to  give  a numerically  equal  voltage  OE%  ; then  OEx  and  OE2 
must  be  in  such  phases  with  respect  to  each  other  as  to  give 

* Art.  161. 


708 


ALTERNATING  CURRENTS 


the  resultant  OR , while  the  components  of  voltage  resolved 
upon  the  current  vector  01  must  be  such  that  the  component 
of  OEx  has  the  same  direction  as  the  current,  and  the  compo- 
nent of  OE g is  in  opposition.  To  obtain  this  relation  the  arma- 
ture voltage  must  fall  back  from  direct  opposition  to  the 
impressed  voltage  by  a space  angle  equal  to  02  + 0V  In  other 
words,  the  armature  conductors  must  take  up  a step  relation 


y 


Fig.  424. — Diagram  showing  Relation  of  Voltages  and  Current  in  a Synchronous 
Motor  for  a given  Field  Excitation  and  Armature  Current.  Impressed  Voltage 
Equal  to  Counter-voltage. 

with  respect  to  the  field  magnetism  that  is  determined  by  the 
vector  values  of  Ev  Ev  and  IZ.  The  work  delivered  to  the 
motor  by  the  circuit  is  IE1  cos  0V  and  that  utilized  by  the  motor 
armature  in  furnishing  power  and  overcoming  the  magnetic  and 
frictional  losses  is  IE2  cos  02  ; while  that  lost,  due  to  armature 
resistance,  is  their  difference,  and  is  equal  to 

I2Z  COS  = OS  X 01. 

It  will  be  seen  that  for  small  loads  on  the  motor  the  current 
with  the  assumed  voltage  may  lead  the  impressed  voltage,  as 
shown  in  Fig.  424,  that  is,  0X  is  negative  ; but  that  as  the  load 
increases  (and  the  length  of  OR  therefore  increases),  the 
motor  counter-voltage  is  caused  to  swing  farther  back  in  step 
compared  with  the  impressed  voltage,  so  that  the  current  takes 
a lagging  position,  as  shown  in  Fig.  425.  The  construction 
indicates  that  the  current  is  always  more  than  180°  in  the  lead 
of  the  counter-voltage,  when  the  impressed  and  counter  vol- 
tages are  equal,  and  that  the  value  of  (#2  + 8 x)  increases  when 


SYNCHRONOUS  MACHINES 


709 


the  load  on  the  motor  is  increased.  The  value  of  the  current 
I is  one  half  greater  in  Fig.  425  than  in  Fig.  424,  the  input  of 
the  motor  IE1  cos  01  is  50  per  cent  greater,  and  the  output 
IE 2 cos  #2  about  45  per  cent  greater.  The  latter  is  increased 
in  a smaller  proportion,  because  the  I2R  loss  increases  directly 
as  I2.  The  construction  of  these  figures  makes  it  manifest  that 
the  parallelogram  of  voltages  OEv  OE2  and  resultant  OR  de- 
termines the  values  of  0l  and  02  so  that  each  has  a particular 
value  for  each  given  value  for  the  impressed  and  counter-vol- 
tages and  the  magnitude  of  the  motor  load. 


Fig.  125. — Vector  Diagram  of  Voltages  and  Current  when  the  Load  is  increased,  hut 
the  Conditions  are  Otherwise  as  in  Fig.  424. 

The  motor  will  continue  to  operate,  as  the  load  is  increased, 
until  &2  has  attained  such  a value  that  E2  cos  02  x I becomes  a 
maximum  ; then,  if  an  additional  load  is  put  on  the  motor,  the 
corresponding  increase  of  02  will  cause  IE2  cos  d2  to  decrease, 
and  the  motor  will  fall  out  of  synchronism  and  stop,  because 
the  maximum  value  of  its  torque  is  not  sufficient  to  pull  the 
load.  In  the  case  under  consideration  (when  the  impressed 
and  counter  voltages  are  equal)  this  will  not  occur  with  well- 
designed  alternators  until  a load  much  above  that  for  which 
the  machine  is  rated  is  reached. 

As  was  stated,  the  counter-voltage  at  which  the  motor  is  run, 
and  therefore  its  excitation,  has  an  important  bearing  upon  the 


710 


ALTERNATING  CURRENTS 


stability  of  operation  and  the  efficiency  of  transmission.  If  the 
motor  voltage  is  made  larger  than  that  of  the  generator,  the 
current  and  voltage  relations  may  be  shown  by  a construction 
similar  to  that  used  in  Fig.  424.  If  OR  in  Fig.  426  represents 
the  magnitude  and  direction  of  the  resultant  voltage,  and  the 
impressed  and  counter  voltages  have  magnitude  OEx  and  OEv 
then  the  parallelogram  can  be  completed  with  the  values  of  the 
angles  6l  and  02  shown  in  the  figure.  In  this  illustration  the 
counter-voltage  is  greater  than  the  impressed  voltage.  Now 


Fig.  426.  — Diagram  showing  the  Effect  of  Variation  of  Excitation  in  a Synchronous 

Motor. 


suppose  the  motor  excitation  is  changed  so  that  the  counter- 
voltage is  OE2 having  the  same  horizontal  projection  as  OEv 
while  OR  and  the  impressed  voltage  have  the  same  values  as 
before ; then  the  phase  relations  are  as  shown  by  the  lines 
OEx\  OR , and  OE2.  The  values  of  OE2  cos  02  and  OE2'  cos  02 
are  equal  by  the  construction,  since  the  points  E2  and  E2  are  in 
the  same  vertical  line  ; and  since  OR  lias  the  same  magnitude 
and  position  in  the  two  cases  the  current  is  the  same.  Also, 
OEl  cos  dj  and  OE cos  0X'  are  equal ; but  in  the  first  instance 
the  counter-voltage  is  greater  than  the  impressed  voltage  and 
the  current  leads  the  impressed  voltage,  and  in  the  second 
instance  the  impressed  voltage  is  the  greater  and  the  current 


SYNCHRONOUS  MACHINES 


711 


lags.  This  construction  shows  (with  a fixed  value  of  impressed 
voltage)  that  for  every  load  on  the  motor,  except  that  correspond- 
ing to  a zero  angle  of  lag,  there  are  two  values  of  the  excitation 
which  cause  the  same  armature  current  to  flow,  the  current  lead- 
ing the  impressed  voltage  with  the  higher  excitation  and  lagging 
by  an  equal  angle  with  the  lower  excitation ; hence  an  over-ex- 
cited motor  acts  upon  the  line  current  very  much  like  a con- 
denser, and  an  under-excited  motor  acts  like  an  inductance  coil. 

Figure  433  shows  the  armature  current  of  a synchronous  motor 
operating  under  fixed  load  on  a circuit  of  fixed  voltage,  plotted 
as  a function  of  the  field-exciting  current.  This  shows  plainly 
that  the  armature  current  assumes  the  same  numerical  value  for 
each  of  two  excitations.  The  vector  diagrams  show  that  the  les- 
ser excitation  corresponds  to  a current  lagging  and  the  greater  ex- 
citation to  a current  leading  with  respect  to  the  impressed  voltage. 

The  excitation  at  which  the  motor  will  do  the  most  work 
with  a given  current  flowing,  or  will  carry  a given  load  with 
the  least  current  (and  hence  do  it  with  the  best  power  factor), 
is,  as  is  proved  later  (Art.  169),  that  which  causes  the  current 
to  come  into  phase  with  the  impressed  voltage.  In  this  case 
the  current  for  a given  load  is  a minimum,  and  E2  cos  02  is  a 
maximum  for  the  conditions  concerned.  If  the  value  of  E2  in 
Fig.  426  is  increased  (by  increasing  the  excitation  of  the 
motor),  either  the  value  of  I and  the  length  of  OR  which  is 
proportional  to  it,  or  the  impressed  voltage,  must  be  increased, 
provided  IE2  cos  02,  which  is  equal  to  the  motor  load,  is  con- 
stant. On  the  other  hand,  taking  the  vector  rectangle  OE2REv 
if  E2  decreases  numerically  while  Ex  remains  numerically  con- 
stant, the  values  of  the  angle  0X  and  the  current  decrease  until 
the  current  and  impressed  voltage  come  into  phase,  after  which 
a further  decrease  of  E2  causes  the  current  to  increase  again 
and  the  angle  of  lag  to  increase  in  the  positive  direction.  All 
this  is  shown  by  the  relations  between  Ev  /,  and  E2  in  the  figure. 

The  value  of  IE2  cos  02  (that  is,  the  load  carried  when  the 
OR 

current  is  , when  Z is  the  synchronous  impedance,  and  the 

Z 

counter-voltage  is  either  OE2  or  OE2~)  is  proportional  to  the 
area  of  the  rectangle  OlqP , since  01  is  proportional  to  I.  Now 
if  the  motor  load  is  kept  constant,  but  its  field  excitation  is 
changed,  the  corner  of  the  corresponding  rectangle  must  move 


715 


ALTERNATING  CURRENTS 


from  q,  but  the  locus  of  the  positions  of  the  corner  is  a hyper- 
bola with  its  origin  at  0 and  the  rectangular  axes  x and  y,  since 
the  rectangles  included  between  the  axes  and  the  ordinate  and 
abscissa  for  each  point  must  be  all  of  equal  area.  Consequently, 
the  point  of  the  vector  representing  E2  at  the  least  current  for 
the  given  load  must  be  in  the  rectangular  hyperbola  mm  which 
passes  through  q.  This  point,  which  is  E2"  for  the  given  load 
and  the  impressed  voltage  Ev  is  found  by  laying  off  the  hori- 
zontalline equal  in  length  to  0E1  which  will  just  reach  between 
the  hyperbola  and  the  line  OR.  This  cuts  OR  at  R' , and  the 


sought  for  minimum  current  is  1= 


9L 

X 


OR' 
Z ’ 


where  X is 


the  synchronous  reactance  and  Z the  synchronous  impedance. 
The  impressed  voltage  and  current  are,  under  these  conditions, 
in  phase  with  each  other.  If  a larger  or  smaller  counter- voltage 
than  OE 2"  is  used,  such  as  0E2  or  OE 2',  the  impressed  voltage 
and  current  are  thrown  out  of  phase,  and  the  current  in  the  cir- 
cuit is,  therefore,  increased,  if  the  impressed  voltage  remains  of 
the  same  numerical  value,  which  causes  increased  PR  losses  and 
armature  reactions.  The  excitation  of  the  motor  which  brings 
the  current  and  impressed  voltage  into  step  for  a given  load 
depends  upon  the  relation  of  the  synchronous  impedance  of  the 
motor  circuit  to  its  resistance.  When  the  angle  i/r  is  60°,  that 
is,  when  Z — 2 R,  the  counter-voltage  is  equal  to  the  impressed 


voltage  at  zero  lag  of  the  current,  since  ^ = :L—  under  con- 
ditions of  maximum  output  (Art.  169)  ; and  when  Z is  greater 
or  less  than  2 R the  counter-voltage  is  respectively  greater  or 
less  than  the  impressed  voltage  at  zero  lag.  Putting  X and  Z 
for  synchronous  reactance  and  impedance,  then  X = V3  R when 
Z = 2 R,  since  Z — VR2  + X2.  In  the  machines  which  are  now 
commonly  built,  the  synchronous  impedance  of  the  armature  cir- 
cuit is  commonly  larger  than  twice  the  resistance,  and  the  max- 
imum power  factor  and  output  are  obtained  in  such  synchronous 
motors  by  an  excitation  which  gives  a counter-voltage  that  is 
greater  than  the  impressed  voltage. 

Figure  427  shows  polar  diagrams  for  various  excitations  at 
which  a small  motor  under  constant  load  was  provided  with 
power  by  a generator  of  exactly  the  same  size  and  tj'pe  as  the 
motor.  Lines  OEv  OEv  and  OR  are  the  generator,  motor,  and 


SYNCHRONOUS  MACHINES 


718 


Fig.  427.  — Vector  Diagrams  obtained  from  Experiments  made  with  Two  Small 
Similar  Machines,  One  Running  as  a Motor,  the  Other  as  a Generator. 


714 


alternating  currents 


resultant  voltages  respectively  acting  in  the  circuit  comprising 
the  two  machines,  and  01  is  the  current.  These  were  obtained 
from  a diminutive  pair  of  single-phase  alternators  connected 
together  and  operated  as  motor  and  generator  in  a classic  ex- 
periment of  Professors  Ryan  and  Bedell.  They  illustrate  the 
vector  relations  of  current  and  voltages  for  motors  of  all  sizes. 
The  angle  by  which  the  current  lags  behind  the  resultant  vol- 
tage was  obtained  from  the  synchronous  impedance  of  the  arma- 
ture circuit.  It  may  be  clearly  seen  from  the  diagrams  that  the 
current  swings  from  a position  of  large  lag  with  reference  to 
the  'generator  voltage  at  a small  excitation  of  the  motor  as  in 
diagrams  Nos.  1,  2,  and  3 ; into  phase  with  it,  as  between  dia- 
grams Nos.  4 and  5,  the  point  of  minimum  current  for  the  given 
load  on  the  motor ; and  finally  into  a position  of  large  lead 
when  the  motor  is  greatly  over-excited,  as  in  diagrams  Nos.  7 
and  8.  The  numerical  values  of  Ev  Ev  /,  and  Ez  and  the  an- 
gular relation  of  E1  and  Ev  as  observed,  are  shown  in  the  figure. 
The  ratio  of  Ez  to  1 is  the  internal  impedance  of  the  motor. 

Figure  428  shows  the  armature  current  for  different  motor 
field  excitations  when  the  motor  was  under  the  same  small  con- 
stant load.  Curves  I),  F , and 
Cr  in  the  same  figure  show 
plainly  the  tendency  of  the 
armature  reactions  and  leak- 
age reactance  to  reduce  differ- 
ences between  numerical  val- 
ues of  motor  and  generator 
voltages.  Curve  D repre- 
sents the  generator  voltage, 
and,  although  the  generator 
excitation  was  constant,  the 
generator  voltage  rises  as 
the  motor  voltage  is  in- 
creased. This  is  due  to  the 
reaction  caused  by  the  cur- 
rent swinging  from  a lag  to 
a lead  with  reference  to  the 
generator  voltage.  At  the 
same  time  the  motor  voltage,  which  is  represented  by  curve  F, 
at  first  is  larger  and  then  grows  smaller  than  would  be  the  case 


MOTOR  FIELD  CURRENT 


Fig.  428.  — Curves  showing  Relation  of  Vol- 
tages and  Armature  Current  for  Constant 
Load  and  Variable  Excitation  in  a Syn- 
chronous Motor. 


SYNCHRONOUS  MACHINES 


715 


were  no  reactions  present.  The  curve  G-  represents  the  motor 
voltage  considering  reactions  absent.  The  effect  in  the  motor 
is  caused,  as  in  the  generator,  by  the  current  changing  its  lead 
with  reference  to  the  motor  voltage. 

This  series  of  experiments  and  the  preceding  discussions  show 
that  there  was  some  foundation  for  the  statement  of  earlier  ex- 
perimenters, that  alternators  must  have  self-inductance  in  their 
armature  circuits  if  they  are  designed  to  be  run  in  parallel. 
The  application  of  that  statement  to  the  case,  however,  is  falla- 
cious, since  alternators  operating  in  parallel  should  require 
much  less  than  the  torque  of  normal  load  to  hold  them  in  step, 
so  that  the  synchronizing  tendency  of  armatures  with  small  in- 
ductance is  ample  to  make  them  run  in  parallel,  and  for  either 
parallel  working  or  for  operation  as  synchronous  motors  a small 
armature  impedance  is  of  the  greatest  importance. 

169.  Locus  Diagrams  of  Currents,  Voltages,  and  Loads  of  the 
Synchronous  Motor.  — In  Fig.  426  a locus  is  given  whereby  the 
motor  excitation  can  be  obtained  when  the  load  is  constant  and 
the  current  varies.  Suppose  now  that  the  current  and  im- 
pressed voltage  are  maintained  of  constant  values,  then  Fig.  429 
results.  The  locus  of  the  impressed  voltage  is  a circle  EXE2E", 
with  its  center  at  0.  The  current  represented  by  01  being  con- 
stant, OR , the  resultant  voltage,  is  constant,  if  it  is  assumed  that 
the  synchronous  impedance  is  constant.  The  latter  assumption 
is  not  exact  for  commercial  machines  on  account  of  the  effect  of 
saturation  in  the  magnetic  circuit  with  different  excitations  of 
the  field  magnet  as  well  as  the  effect  of  changes  of  the  power 
factor  of  the  motor,  but  the  assumption  may  prevail  for  the  pur- 
pose of  the  following  exposition.  Then  in  order  that  the  im- 
pressed and  counter-voltages  may  always  combine  to  equal 
OR  the  locus  of  the  counter-voltage  E%  must  be  the  circle 
E2E2'E2"  with  its  center  at  R and  its  radius  equal  to  Ev  The 
geometrical  construction  of  the  figure  shows  that  these  relations 
are  necessary  to  fulfill  the  condition  of  constant  current  and  im- 
pressed voltage.  When  the  excitation  gives  the  counter-voltage 
OEv  the  construction  shows  that  the  mechanical  load  is  OT  x 
/0(=  IE2  cos  d2,  where  d2  is  the  angle  between  0E2  and  _Z49), 
which  will  be  assumed  to  be  the  amount  of  power  required  to 
drive  the  armature  at  synchronism  without  external  load.  The 
current  01  leads  OEx  by  a large  angle  so  that  there  is  a large 


716 


ALTERNATING  CURRENTS 


leading  reactive  component  of  current  equal  to  VI,  while  the 
active  component  of  0EX  QEX  cos  6V  where  0X  is  the  angle  between 
Ex  and  Ix)  is  equal  to  OW,  and  0 W x 01  is  in  this  instance  only 
sufficient  power  to  furnish  the  I2R  losses  and  the  power  re- 
quired to  overcome  magnetic  and  friction  losses.  The  output 
of  the  motor  is  a maximum  when  E2  equals  0E2'  (and  is  equal 


Fig.  429. — Locus  of  Counter- voltage  in  a Synchronous  Motor  when  the  Current  and 
Impressed  Voltage  are  Constant.  The  Current  Vector  is  made  the  Initial  Line. 

to  OT'  x I O')  for  at  that  value  of  E2  the  active  component  of 
the  counter-voltage  is  a maximum,  the  vertical  line  E2  T'  be- 
ing tangent  to  the  arc  E2E2E2  . The  construction  shows 
that  with  this  excitation  of  the  motor,  the  power  is  taken  from 
the  circuit  at  unity  power  factor.  This  obviously  must  be  true 
under  the  conditions,  since  a maximum  output  with  fixed  losses 
predicates  a maximum  input.  As  a similar  construction  may 
be  made  with  the  current  1 of  any  fixed  value,  the  motor  ex- 


SYNCHRONOUS  MACHINES 


717 


citation  that  gives  unity  power  factor  at  any  current  gives 
maximum  output  for  that  current.  The  minimum  excitation 
which  will  keep  the  motor  running  is  when  the  counter-voltage 
equals  OE 2" . At  this  point  the  power  converted  by  the  arma- 
ture is  again  equal  to  OT  x 10  and  is  just  sufficient  to  drive  the 
machine  without  external  load,  while  the  impressed  voltage  OE  " 
again  supplies  an  active  component  OW,  and  OIF  x 01  equals 
the  power  input  required  to  supply  all  losses.  The  current  now 
has  a heavy  lagging  reactive  component  TJI.  The  motor  cannot 
operate  with  the  current  01,  even  without  an  external  load,  if  the 
excitation  is  either  increased  or  decreased  beyond  the  limits  of 
0E2  and  OE2  and  the  impressed  voltage  is  maintained  constant; 
hut  it  must  be  remembered  that  the  armature  reactions  may  con- 
tribute to  the  excitation,  and  a motor,  without  external  load, 
may,  therefore,  run  synchronously,  even  when  not  provided  with 
external  direct  current  excitation.  If,  however,  the  motor  is 
coupled  to  a prime  mover  such  as  a steam  engine,  the  counter- 
voltage may  take  any  position  on  the  complete  circle  of  which 
E2E2  E2  is  an  arc.  In  this  case  when  the  motor  voltage  is  near 
the  top  or  bottom  of  the  circle  locus  so  that  its  active  component 
is  less  than  is  needed  to  provide  the  load  torque,  it  will  draw 
power  partly  from  the  electric  mains  and  partly  from  the  engine. 
When  the  active  component  is  on  the  right  of  0 and  equal  to 
or  greater  than  — OT  the  machine  becomes  a generator  work- 
ing in  parallel  with  the  mains.  The  voltage  locus  of  which 
EXE^E"  is  an  arc  may  also  be  drawn  as  a full  circle;  and  when 
a voltage  vector  is  to  the  right  of  0 power  is  delivered,  but 
when  the  vector  is  to  the  left  of  0 power  is  absorbed.  The 
arc  MM  in  the  figure  is  the  locus  of  E2  when  I equals  ^ 01 ; 
M'M'  is  the  locus  when  I equals  % 01;  and  31" M"  is  the  locus 
in  the  limiting  case  which  might  occur  if  the  armature  reactions 
could  be  negligible  and  the  motor  losses  could  be  exactly  sup- 
plied mechanically  through  the  motor  pulley,  in  which  case  El 
and  E2  would  have  to  always  be  equal  and  opposite. 

A more  generally  useful  locus  is  that  of  the  current  when 
the  impressed  and  counter  voltages  are  maintained  constant 
and  the  load  is  varied.  The  construction  is  shown  in  Fig. 
430,  in  which  OEx  is  the  fixed  vector  of  impressed  voltage  Ev 
Consider  first  the  case  where  the  counter-voltage  E2  equals 
Ev  Then  when  the  counter-voltage  is  in  the  position  OJ 


718 


ALTERNATING  CURRENTS 


opposite  to  0EV  no  current  can  flow  as  there  is  no  resultant 
voltage.  This  position  is  imaginary,  as  the  machine  will  not 
operate  unless  supplied  with  at  least  enough  power  to  overcome 
the  losses.  Assume  now  that  the  motor  armature,  and  hence  Ev 
falls  back  in  step  to  the  position  0E2,  because  of  the  applica- 
tion of  load.  The  resultant  voltage  is  then  OR,  and  the  cur- 

It 

rent  is  01  lagging  behind  OR  by  an  angle  = cos-1—,  where 

Z 

R is  the  armature  resistance  and  Z the  synchronous  impedance. 
The  current  is  1=  When  the  point  E2  moves  along  the 


Therefore,  since  the  current  maintains  a constant  angle  i/r  with 
OR  and  varies  in  length  in  proportion  to  OR,  the  point  I de- 
scribes a circular  locus  0EX  GrB  drawn  on  a diameter  OAT) . which 
makes  an  angle  equal  to  -v/r  with  the  line  0EV  The  diameter 

OB 

of  this  circle  is  equal  to  — — , that  is,  the  diameter  of  the  locus 

z 

of  R divided  by  the  synchronous  impedance.  The  diameter 
OB  is  the  theoretical  value  of  OR  that  would  exist  under  the 


SYNCHRONOUS  MACHINES 


719 


hypothetical  condition  of  OE, 2 in  consonance  with  OFx  and  the 
motor  on  the  circuit  with  its  induced  voltage  in  series  relation 
with  the  impressed  voltage,  that  is  OB  = Ex  + E2.  The  angle 
DOB  equals  i/r,  as  Z is  considered  constant.  The  diameter  of 
the  circle  ODD  equals 

Ex  + E2_2E2_2EX 

z ~ z ' 

since  E2  — Ex  by  the  premises;  and  it  makes  an  angle  with  OEx 
which  is 

Y = cos  1—- 

Now  suppose  that  E2  is  made  OE2,  which  is  less  than  Ex  by 
such  an  amount  that,  when  the  two  are  in  direct  opposition, 
the  resultant  voltage  OR'  in  phase  with  OEx  drives  the  cur- 
rent 00  through  the  armature.  By  the  geometrical  construc- 
tion it  is  seen  that,  when  0E2'  lags  behind  opposition  to  OEv 
OR'  describes  the  circular  locus  R'LJY,  and  therefore,  the  cur- 
rent which,  as  before,  must  maintain  a constant  angular  position 
with  respect  to  the  resultant  voltage  and  be  proportional  to  its 
length,  must  describe  the  circular  arc  OD'P.  The  diameter  of 
this  circle  is 

R'N  _ 2 E2 
Z Z 

Again,  suppose  E2  is  made  greater  than  Ex  and  is  given  the 
value  shown  by  the  line  0E2" . If  E2"  takes  the  position  0F2\ 
shown  in  the  figure,  the  resultant  OR"  drives  the  current  OF 
through  the  synchronous  impedance.  For  this  position  of  0F2" 
the  machine  must  evidently  be  driven  as  a generator,  as  it  is 
driving' current  against  the  impressed  voltage  Fv  It  will  not 
run  as  a motor  until  0E2"  has  fallen  back  sufficiently  to  cause 
the  current  vector  to  take  an  angle  of  less  then  90°  from  the  im- 
pressed voltage  and  have  an  active  component  sufficient  to  sup- 
ply not  less  than  the  losses  of  the  machine,  other  than  field  losses. 
The  locus  of  the  resultant  voltage  OR"  must  by  geometrical 
construction,  as  before,  be  on  the  semicircle  R"ST , and  the  cur- 
rent locus  in  accord  with  the  reasoning  for  the  previous  cases 
is  FD"H.  By  continuing  this  method  the  current  loci  for  any 
desirable  number  of  counter-voltages  may  be  laid  off  with  the 
respective  loci  of  counter  and  resultant  voltages. 


720 


ALTERNATING  CURRENTS 


The  input  can  be  obtained  from  the  product  of  current, 
impressed  voltage,  and  the  cosine  of  the  angle  of  lag  between 
the  two.  The  maximum  current  input  for  the  machine  when 
running  as  a motor  with  minimum  excitation  under  which  the 
motor  will  run  is  slightly  less  than  current  OA  which  equals 

The  maximum  possible  current  input  for  the  machine 


when  running  as  a motor  is  when  there  is  maximum  excitation 
under  which  the  motor  will  run,  and  this  can  then  be  more  than 


current  OD , and  therefore  more  than 


2 E, 


The  mechanical 


load  including  core  losses  equals  the  input  minus  the  armature 
copper  losses,  or  the  total  power  transferred  from  the  mains  to 
cause  rotation  equals  the  impressed  voltage  times  the  active 
component  of  current  minus  the  I2R  losses,  which  can  be  read- 
ily calculated.  Therefore,  we  have  the  input, 

Pi  — EXI cos  0X ; 

and  the  total  transfer  of  electrical  energy  by  the  armature  into 
mechanical  energy  and  core  losses, 

P0  — E2I cos  d2  = EXI  cos  01  — T2R, 


where  d ^ and  02  are  respectively  the  angles  between  impressed 
and  counter-voltages  and  the  current.  The  mechanical  power 
given  out  is,  as  intimated,  less  than  P0  by  an  amount  equal  to 
the  mechanical  and  magnetic  power  losses  caused  by  rotation. 

In  order  to  determine  the  current  that  is  required  at  a given 
power  factor  to  supply  a specific  armature  power  P0 , a locus  of 
currents  for  constant  power  may  be  constructed  in  accordance 
with  the  foregoing  principles.  Thus,  in  Fig.  431  let  OEx  be 
the  direction  of  the  impressed  voltage,  which  is  also  made  the 
initial  line.  Assume  now  that  a certain  output  P0,  including 
rotative  magnetic  and  frictional  losses,  is  being  consumed  by 
the  rotor  or  delivered  from  its  shaft.  Then  the  input  is 
EXI cos  dj  = Pa  + P2R, 

where  E1  is  the  impressed  voltage,  I the  current  flowing,  dj 
the  angle  of  lag  between  the  current  and  impressed  voltage, 
and  R the  resistance  of  the  armature.  From  this: 


P- 


EJcos  d1  _ 


Po 

R' 


R 


SYNCHRONOUS  MACHINES 


721 


and 


t-2  Ex  cos  d,  j 1 E cos2  0X  _ 1 Ex 2 cos2  d1  P0 
R + 4 i?2  ~ 4 R2  R' 


lienee, 


1 Ei  cos  $i  ll  Ex2  cos2#,  Ra 

2 R M ^ if 


(1) 


From  this  expression  for  I when  the  armature  power  is  P0  it  is 
seen  that  there  are  two  values  of  armature  current  for  each 


Fig.  431. — Diagram  showing  Current  Loci  in  a Synchronous  Motor  for  Constant 
Power  when  the  Impressed  Voltage  is  Constant. 


value  of  cos  6X  except  one,  and  also  that  the  currents  are  nu- 
merically equal  but  of  opposite  algebraic  signs  for  equal  posi- 
tive and  negative  values  of  cos  0X.  The  values  of  the  currents 

...  • , i P0  i E, 2 COS  2 0X  rp 

contain  imaginary  terms  when  lo  more 

R R2 

clearly  observe  these  conditions  the  locus  of  constant  power,  P0, 
3a 


722 


ALTERNATING  CURRENTS 


HMB  is  drawn.  The  locus  is  a circle  having  a radius  equal  to 


when  P0  and  6X  equal  zero.  Then  the  current  is  in  phase  with 
the  voltage  and  flows  through  the  windings  of  the  armature  in 
accordance  with  Ohm’s  law  and  all  the  energy  is  utilized  in  I2R 
loss.  Such  a condition  could  only  exist,  of  course,  when  the 
rotor  has  its  rotative  losses  supplied  from  external  mechanical 
sources,  and  has  swung  to  such  a position  that  the  counter- 
voltage forms  a resultant  with  the  impressed  voltage  of  such 
phase  angle  that  the  current  is  in  consonance  with  the  latter. 
These  conditions  are  only  fulfilled  when  the  counter-voltage  is 
at  right  angles  with  the  current  and  of  definite  length.  This 

current  ^ is  laid  off  at  00  in  the  Fig.  431.  When  01  only  is 
R 

zero,  the  formula  for  current  becomes 


The  larger  value  obtained  by  this  formula  is  laid  off  as  OB  and 
the  smaller  value  as  OH  in  the  figure.  The  distances  OH  and 
BO  are  evidently  equal. 

Now  take  any  value  of  current  I obtained  for  a given  Pa  and 
for  any  angle.  Assume  that  01  at  lag  angle  0X  is  such  a cur- 
rent. Then  draw  the  perpendicular  I>K  from  a point  I),  mid- 
way between  0 and  C or  H and  B,  and  connect  the  points  D 


E 

From  the  last  formula  it  is  seen  that  the  current  I equals  — 1 

R 


and  I.  The  length  01)  = = ^ 00,  and  by  construction 


and 


Also 


2 R 


By  construction  (2)F)2  = ( PK)2+  (FT/)2. 


SYNCHRONOUS  MACHINES 


723 


Substituting  in  this  the  values  of  PK  and  KI  and  using  the 
value  for  current  given  in  the  formula  for  current  in  Equation 
(1),  then 


E*n*-psfvi  -oo^' 


+ 


1 Ex  cos  0, 
R 


± 


ll  E2  cos2  6>,  P0 

V4  R2  R 


which  reduces  to 


1 p 2 

(Diy  =-^y- 
K J 4 R2 


Po 

R 


(2) 


or 


pi=hp  = ±JI*1-^ 

'4  R2  R 


(3) 


Since  I is  assumed  in  the  premises  to  he  the  current  which  can 
supply  the  power  P0  at  a given  value  for  the  angle  6V  the  point 
I so  long  as  P0  is  constant  must  travel  the  locus  HMBR  which 


must  be  a circle  having  a radius  equal  to 


El 

R2 


P 

—2,  with  its 

R 


1 E, 


center  P at  a distance  equal  to  - ^ from  the  origin  0. 

2 R 


Any 


power  P0  was  assumed,  hence  by  assuming  a series  of  outputs, 
the  circular  loci  for  the  outputs  are  concentric  circles  such  as 
those  marked  M,  M\  and  M" . The  largest  locus  possible  has  a 
radius  about  equal  to  PO , P0  then  being  only  sufficient  to  supply 
the  no-load  rotative  losses.  The  smallest  is  the  point  P , where 
the  radius  is  zero,  the  power  factor  is  unity  and  the  greatest  out- 
put occurs.  For  this  locus  there  is  only  one  current  OP , half 
the  input  power  is  used  up  in  I2R  loss  and  the  other  half  in 

turning  the  armature  since  — = - and  I=~^.  The 
s R 4 R2  2 R 

efficiency  is  thus  less  than  50  per  cent.  Commercial  synchronous 
motors  could  not  approach  this  amount  of  load  without  causing 
excessive  and  disastrous  heating.  The  maximum  efficiency  oc- 
curs when  the  copper  losses  equal  the  rotation  losses,  which 
may  usually  be  assumed  without  serious  error  to  be  of  fixed 
value.  By  projecting  01  upon  the  voltage  line  OEx  and  at  right 
angles,  the  active  and  quadrature  components  of  the'  current, 
OF  and  0(7,  are  obtained.  By  suitably  changing  the  scale  of 


724 


ALTERNATING  CURRENTS 


the  figure,  01,  OF,  and  OGr , may  be  read  off  directly  as  total 
kilovolt-amperes,  kilowatts,  and  quadrature  volt-amperes  re- 
spectively. The  output  may  be  determined  for  any  locus  by 
marking  the  locus  with  the  power  assumed  in  its  construction. 
Since  the  radii  of  the  current  loci  are  scaled  in  amperes  so  that 


FI 

4 R2 


Po 

R' 


the  radius  of  the  locus  circle  depends  upon  the  output,  as  Fx 
and  R are  constant.  Therefore,  loci  for  various  outputs  can 
be  readily  laid  out  by  making  the  radius 


r — \l A — — ° 

' R 

IE2 

where  A is  a constant  equal  to  - — -L. 

4 R2 


By  combining  Figs.  430 


and  431  all  of  the  information  desired  concerning  the  motor 
voltages  and  phase  angles  is  obtained  for  any  output  when  the 
voltage  and  power  factor  of  the  input  are  known.  A figure  or 
construction  similar  to  that  of  Fig.  410  may  be  made  for  a 
synchronous  motor  having  constant  power  output,  variable 
induced  voltage,  and  constant  impressed  voltage.  By  a process 
of  reasoning  similar  to  that  used  in  discussing  the  division  of 
load  in  alternators,*  it  may  be  shown  that  the  induced  voltage 
locus  (constant  power  circle)  is  a circle  having  a radius 

JA 

and  the  polar  coordinates  E0  and  — 1 - ■ , 
1 2 2 cos  6a 

where  P0  is  the  motor  output,  F1  the  impressed  voltage,  E2  the 
motor  counter  voltage,  and  6a  the  internal  angle  of  lag.  The 
angle  included  between  the  coordinates  is  (#a  — /3).  The  con- 
struction is  like  that  of  Fig.  410,  except  that  the  angle  O'OB  is 
(0O  — /3)  instead  of  (#a  + /3)  and  the  locus  radius  is  shorter 
than  00',  which  throws  the  motor  counter-voltage  to  the  right 
of  the  terminal  voltage  OB' and  causes  the  loci  TT,  etc.,  to  be 
at  the  right  of  OV. 

170.  Curve  showing  the  Relation  of  Armature  Current  to 
Excitation.  — The  relation  of  armature  current  to  field  excita- 
tion may  be  plotted  for  a motor  operating  under  the  conditions 
considered  above,  namely,  when  driving  a constant  load,  by 

* Art.  162. 


f F2  P0Za 

4 cos2  6.  cos  6. 


SYNCHRONOUS  MACHINES 


725 


taking  the  corresponding  values  of  El  and  I from  a chart  made 
like  Fig.  426,  or  like  Fig.  431.  This  gives  a curve  like  Fig. 
432,  which  has  two  values  of  its  abscissas  for  every  value  of  the 
ordinate  except  the  lowest  and  highest,  points  A'  and  O'  in  the 
figure ; or  for  each  value  of  the  armature  current  there  may 
be  two  values  of  the  excitation,  one  giving  a counter- voltage 


FIELD  EXCITATION 


Fig.  432. — Calculated  V-Curve  for  a Synchronous  Motor,  showing  the  Relation 
between  Field  Excitation  and  Armature  Current  for  a Constant  Load. 

greater  and  the  other  a counter-voltage  less  than  the  impressed 
voltage,  except  at  the  points  of  minimum  and  maximum  arma- 
ture current,  which  correspond  to  but  one  excitation,  as  has 
already  been  explained.  Likewise  there  are  two  values  of  the 
armature  current  for  each  value  of  the  field  excitation  except 
for  the  minimum  and  maximum  excitations  at  which  the  given 


726 


ALTERNATING  CURRENTS 


load  can  be  carried,  namely  at  tlie  points  I)  and  I)'  in  the  figure. 
The  way  in  which  Figs.  426  and  431  are  constructed  shows 
that  the  smaller  the  angle  -*/<-,  or  the  smaller  the  armature  self- 
inductance, the  less  will  be  the  difference  in  the  two  excita- 
tions corresponding  to  any  armature  current ; and  hence  the 

curve  showing  the  relation 
of  excitation  to  current  in  a 
machine  having  a large  re- 
actance compared  with  the 
resistance  is  broad  and 
rounded,  but  the  curve  for 
an  armature  having  a small 
reactance  compared  with  the 
resistance  is  sharp  and  nar- 
row. In  actual  working, 
only  the  lower  part  of  the 
closed  curve  of  Fig.  432  is 
available  for  use.  This  cor- 
responds to  curve  C of  Fig. 
428  and  the  curve  shown  in 
Fier.  433.  These  curves  are 
ordinarily  called  V-curves  on 
account  of  their  shape. 
Curves  for  loads  smaller 
than  the  one  for  which  the 
, „ - ,r  „ curve  is  shown  in  the  figure 

Fig.  433.  — V-Curve  of  a Synchronous  Motor  ^ ° 

Plotted  from  the  Observed  Value  of  Arina-  6Xp<Uld  anti  fall  Outside  ol 
ture  Current  and  Excitation  when  operating  curve,  while  for  larger 

under  Constant  Load.  , , , . , n V .. 

loads  they  contract  and  fall 
within  it.  At  maximum  load,  represented  by  point  D.  Fig. 
431,  the  curve  reduces  to  the  point  P (Fig.  432). 

In  Fig.  430  it  was  seen  that  the  counter-voltage,  which  is 
proportional  to  the  field  flux,  is  also  proportional  to  the  radius 
of  the  current  locus.  The  latter  is  numerically  equal  to 


10  12  u 

EXCITATION 


= OA  when  P,  = Ev 

z 


Hence  if  Z,  the  synchronous  imped- 


ance, has  been  determined,  the  counter-voltage  for  any  current 
locus  is  E2ccr  Z , where  r is  the  radius  of  the  current  locus 
under  consideration.  By  drawing  the  line  OA  in  Fig.  431  so 
that  it  makes  the  angle  y]r  with  the  line  OEv  it  is  possible  to 


' SYNCHRONOUS  MACHINES 


727 


determine  a plot  for  the  V-curve  very  readily  for  any  output  P0. 
Considering  the  power  locus  H31B I to  be  for  the  load  P0  (Fig. 
431),  the  current  01  laid  out  as  an  ordinate  in  Fig.  432  has  an 
abscissa  proportional  to  AI,  the  radius  of  the  current  locus  which 
passes  through  I in  Fig.  430.  If  the  field  reluctance  is  constant 
and  field  reactions  are  included  as  part  of  the  synchronous  im- 
pedance, the  armature  current  locus  may  be  obtained  by  drawing 
an  arc  with  the  radius  01  in  Fig.  431.  Where  it  cuts  the  con- 
stant power  output  locus  HMBI  a second  time  at  Ix  the  excita- 
tion is  proportional  to  the  distance  between  A and  Iv  In  Fig. 


_27  _27 

431  it  is  seen  that  as  0 C = and,  in  Fig.  430,  OA  = , there- 

ZJ 


R 


_z? 

fore  since  BOA,  Fig.  431,  equals  y\r,  OA  — — i in  Fig.  431  is  the 

zj 


horizontal  side  of  the  right-angled  triangle  OA  C.  The  curve 
of  Fig.  432  is  constructed  from  Fig.  431  as  follows:  the  points 
on  the  curve  whose  ordinates  are  equal  to  currents  01  and  OIv 
Fig.  431,  are  Ix  and  B'  in  Fig.  432,  with  abscissas  equal  to  AI 
and  AI'  of  Fig.  431,  while  the  current  OJ  is  represented  by  the 
ordinate  of  point  J with  abscissa  equal  to  AJ  of  Fig.  431.  The 
current  01  (Fig.  431)  is  so  large  compared  with  OB , which 
represents  the  current  when  the  armature  IB  drop  is  equal  to 
I Bv  that  the  efficiency  would  be  very  low.  The  useful  part  of 
the  curve  of  Fig.  432  is,  therefore,  only  a short  distance  on  either 
side  of  A',  the  ordinate  and  abscissa  of  which  are  equal  to  OH  and 
AH  of  Fig-.  431.  The  maximum  load  for  any  excitation  may 
readily  be  determined  from  Fig.  431  ; thus,  if  the  excitation  is 
proportional  to  AS,  the  largest  load  which  can  be  carried  is  rep- 
resented by  the  power  locus  Mr,  as  this  is  the  smallest  power 
locus  that  is  cut  by  the  current  locus  SS'  having  .4$  as  a radius. 

It  must  not  be  supposed  that  the  armature  voltages  actually 
rise  in  proportion  to  the  amperes  of  field  excitation,  even  for 
constant  reluctances,  as  indicated  by  the  abscissas  of  Fig.  432, 
because  the  quadrature  current  when  the  excitation  is  less  than 
AH  (Fig.  432)  strengthens  the  fields,  and  when  greater  than  AH 
weakens  the  fields;  but  this  quadrature  current  calls  for  a pro- 
portional decrease  or  increase  of  field  current  in  the  same  amount 
that  would  be  called  for  if  the  reactions  were  zero  and  the  coun- 
ter-voltages were  proportional  to  the  abscissas  in  the  figure.* 


* McAllister’s  Alternating  Current  Motors,  p.  195. 


728 


ALTERNATING  CURRENTS 


171.  Application  of  the  Parallel  Operation  and  Synchronous 
Motor  Diagrams  to  Machines  Wound  with  Any  Number  of 
Phases.  — The  synchronous  motor  diagrams  given  in  the  pre- 
vious articles  are  suitable  for  use  for  any  single  or  polyphase 
machines.  In  the  case  of  single-phase  machines  the  power  in- 
put, voltage,  and  current  used  in  the  diagrams  are  those  meas- 
ured in  the  supply  mains  by  wattmeter,  voltmeter,  and  am- 
peremeter. For  quarter-phase  machines  one  full  phase  may  be 
conveniently  considered  using  its  voltage,  current,  and  power 
as  in  the  single-phase  machine.  The  powers  must  then  be 
doubled  to  obtain  the  values  of  motor  input  and  output.  Tri- 
phase machines  may  be  dealt  with  in  the  same  manner,  using 
the  voltage,  current,  and  power  in  one  branch  for  obtaining  the 
diagrams  and  then  multiplying  the  inputs  and  outputs  by  three. 
When  the  tri-pliase  motor  is  wye  connected,  the  branch  voltage 

is  the  line  voltage,  and  when  the  motor  is  delta  connected 

V3 

the  branch  current  is  — the  line  current.  Six-phase  mesli- 

V3 

connected  motors  may  be  treated  in  the  same  way  as  similarly 
connected  tri-phase  machines,  except  that  the  power  shown  by 
the  single-phase  diagrams  must  be  multiplied  by  six  to  get  the 
entire  power.  For  six-phase  machines  with  a star  connection, 
the  voltage,  current,  and  power  per  phase  are  the  same  as  in 
the  mesh  connection. 

Prob.  1.  Given  a three-phase  wye-connected  synchronous 
motor  operating  under  2200  volts  impressed  voltage  at  its  ter- 
minals with  a frequency  of  60  periods  per  second.  The  syn- 
chronous impedance  per  phase,  when  the  frequency  is  normal, 
is  considered  constant  at  a value  of  5 ohms  and  is  composed 
of  such  proportions  of  resistance  and  reactance  that  the  current 
lags  75°  behind  the  resultant  voltage.  Giving  the  vector  of 
impressed  voltage  a fixed  position,  draw  the  current  loci  for 
100,  200,  and  300  amperes ; the  constant  total  armature  me- 
chanical power  loci  for  100,  200,  and  400  horse  power,  assum- 
ing that  the  mechanical  and  magnetic  losses  are  constant  at  12 
horse  power ; and  the  V-curves  for  the  latter  powers.  Also, 
giving  the  current  vector  a fixed  position,  draw  the  impressed 
voltage  and  counter- voltage  loci  for  currents  of  100,  200.  and 
300  amperes.  Complete  the  locus  circles  in  the  latter  diagrams 


SYNCHRONOUS  MACHINES 


729 


and  mark  the  positions  where  the  machine  is  a generator  work 
ing  in  parallel  with  the  supply  mains. 

Prob.  2.  Draw  the  V-curves  for  the  generator  and  conditions 
of  Prob.  1,  Art.  162,  by  the  method  given  in  that  Article,  when 
the  total  powers  are  100,  50,  and  25  kilowatts. 

172.  Methods  of  testing  Alternators.  — The  general  defini- 
tions of  efficiencies,  which  have  already  been  explained  with 
regard  to  transformers,*  are  applicable  to  alternators,  although 
in  the  case  of  alternators  the  power  received  by  the  field  coils 
is  usually  supplied  from  a direct  current  exciter  circuit.  The 
principal  causes  of  loss  are  the  same  as  those  in  direct-current 
machines ; but  on  account  of  increased  frequencies  of  the  mag- 
netic cycles  in  the  armature  cores,  the  effects  of  eddy  currents 
and  hysteresis  are  intensified.  On  this  account  particular  care 
is  required  in  selecting  and  annealing  the  iron  for  the  armature 
cores,  and  in  insulating  the  armature  disks  from  each  other. 
Advantage  is  also  taken  of  every  opportunity  for  ventilation. 

The  conditions  under  which  a test  of  alternator  efficiency  is 
to  be  executed  are  usually  specified  with  great  care,  including 
the  speed,  the  load,  the  voltage,  the  current,  the  rise  of  tem- 
perature of  windings,  etc.,  corresponding  to  the  circumstances 
under  which  it  is  desired  to  have  the  machine  work.  Field 
exciting  current,  rise  of  temperature  of  the  bearings  and  the 
collector  rings,  and  other  phenomena  of  the  operation  of  the 
machine  are  also  ordinarily  observed  and  recorded.  The  con- 
ditions for  making  standard  tests  are  given  in  the  Standardiza- 
tion Rules  of  the  American  Institute  of  Electrical  Engineers. 

1.  Efficiency  by  Rated  Motor  and  Stray  Power  Measure- 
ments. — In  case  a transmission  dynamometer  is  used  to  measure 
the  power  absorbed  by  an  alternator,  the  commercial  efficiency  is 
equal  to  the  electrical  output  shown  by  a wattmeter  divided  by 
the  input  shown  by  the  dynamometer  readings  plus  the  power 
used  for  separately  exciting  the  field  magnet.  When  the  ma- 
chine is  separately  excited,  as  is  the  case  in  standard,  modern, 
commercial  alternators,  the  energy  supplied  to  the  field  mag- 
net may  be  measured  by  amperemeter  and  voltmeter  or  by 
wattmeter.  The  machine  friction  may  be  determined  from 
the  dynamometer  readings  when  the  machine  is  run  with  its 
field  magnet  unexcited  and  the  external  circuit  open.  The 


* Art.  137. 


730 


ALTERNATING  CURRENTS 


loss  due  to  hysteresis  and  eddy  currents  in  the  armature  core, 
armature  conductors,  and  pole  faces  may  be  approximately 
determined  from  the  dynamometer  readings  when  the  machine 
is  operated  with  its  field  magnet  separately  excited  and  the 
external  circuit  open,  and  the  total  loss  may  be  approximately 
separated  into  its  component  parts  by  substituting  in  the  for- 
mulas * previously  given,  the  power  being  indicated  from  the 
dynamometer  readings  for  two  values  of  field  excitation.  The 
I2R  losses  in  field  and  armature  conductors  may  be  readily  com- 
puted from  the  respective  hot  resistances.  In  the  special  case 
where  the  machine  is  self-excited  by  a rectified  current,  the 
field  current  will  be  less  than  that  calculated  from  the  voltage 
and  resistance.  In  that  case  it  is  advisable  to  use  an  alternat- 
ing-current amperemeter,  such  as  an  electrodynamometer,  to 
determine  the  effective  current.  From  this  the  I2R  loss  may 
be  computed  if  the  hot  field  resistance  is  known  ; or,  the  field 
loss  may  be  directly  determined  with  more  assurance  of 
accuracy  by  a wattmeter  in  the  field  circuit. 

Instead  of  a tnechanical  transmission  dynamometer,  which  is 
generally  of  a low'  degree  of  accuracy  even  if  available,  it  is 
convenient  to  mechanically  connect  the  alternator  under  test 
to  an  electric  motor,  of  which  the  no-load  losses  and  curve  of 
efficiency  as  a function  of  the  load  are  known.  The  input  of 
the  alternator  under  test  can  then  be  calculated  from  the  elec- 
trical input  of  the  machine  that  is  used  as  a driving  motor 
by  deducting  its  losses.  A machine  used  in  this  way  as  the 
driving  motor  is  commonly  said  to  be  “ rated  ” when  the  rela- 
tions between  its  outputs  and  inputs  are  known  for  all  loads 
and  a number  of  speeds.  It  is  desirable  to  connect  the  rated 
motor  by  a flexible  coupling  to  the  alternator  under  test. 
When  the  motor  is  connected  to  the  test  machine  by  belt,  the 
loss  of  energy  caused  by  the  belt  is  included  in  the  power 
delivered  by  the  motor.  This  loss  may  be  approximately  deter- 
mined by  running  the  test  alternator  as  a synchronous  motor 
with  and  without  the  belt  and  deducting  from  the  difference 
of  the  input  readings  the  friction  and  windage  losses  of  the 
rated  driving  machine.  During  this  test  the  rated  machine 
should  be  disconnected  from  the  supply  mains  and  be  un- 
excited. 


* Art.  110. 


SYNCHRONOUS  MACHINES 


731 


One  of  the  methods  of  obtaining  frictional  or  fixed  losses  is 
sometimes  called  the  retardation  test.  The  machine  under  test 
may  be  brought  to  full  speed  and  then  have  the  driving  force 
removed.  The  speed  is  then  observed  at  certain  intervals  of 
time.  The  friction  loss  is  determined  from  the  formula, 

p 2 7r2  k r r,2  — r,2i 

T 

where  P is  the  power  in  watts  dissipated  in  friction,  Vx  and  T r2 
are  the  speeds  in  revolutions  per  second  at  the  beginning  and 
end  of  the  time  interval  of  T seconds,  and  K is  the  moment  of 
inertia  in  kilogram-(meters)  2 of  the  rotating  part.  By  exciting 
the  field  magnet,  core  losses  plus  friction  may  be  approxi- 
mately obtained  in  this  way.  This  method  requires  that  the 
weight  and  dimensions  of  the  rotating  part  be  known  in  order 
that  K may  be  determined,  and  even  at  the  best  is  not  very 
accurate  for  ordinary  dynamo  tests. 

It  is  usually  best  to  measure  the  losses  under  conditions  of 
no  load  operation  and  compute  the  efficiency  by  taking  the 
ratio  of  the  output  under  consideration  to  the  output  plus  the 
losses,  rather  than  to  measure  the  input  and  output  and  take 
the  ratio  of  their  measured  values.  Windage,  friction,  and 
core  losses  at  no  load,  the  field  magnet  being  excited  with  the 
normal  excitation  for  the  load  concerned  minus  the  armature 
back  turns  for  that  load,  may  be  measured  by  means  of  the 
rated  motor,  and  the  PR  losses  may  be  computed  after  the  hot 
resistance  of  the  windings  has  been  measured.  A correction 
must  then  be  made  for  the  effect  of  armature  reactions  on  the 
distribution  and  density  of  magnetism  in  the  armature  core 
which  alter  the  core  losses  and  also  the  eddy  current  and  hys- 
teresis losses  in  the  pole  faces  of  the  field  magnet.  The  magni- 
tude of  this  correction  depends  upon  the  type  of  machine,  but  it 
ought  not  to  be  large  in  well  designed  and  constructed  machines. 
To  obtain  the  armature  resistance  per  phase  of  a wye-connected 
tri-phaser,  the  measurement  may  be  made  from  each  line  termi- 
nal to  the  neutral  point.  Twice  the  resistance  is  obtained  by 
making  the  measurement  from  one  line  terminal  to  another. 
For  a delta-connected  tri-phaser,  two  thirds  of  the  resistance 
per  phase  is  obtained  by  measuring  from  one  line  terminal  to 
another. 


732 


ALTERNATING  CURRENTS 


The  load  given  out  by  the  alternator  under  test  can  some- 
times be  utilized,  at  least  in  part,  by  being  delivered  to  the 
service  mains  of  the  plant  in  which  the  test  is  being  made. 
However,  frequently  all  and  usually  part  of  the  load  must  be 
absorbed  in  metal  or  water  rheostats. 

A test  of  only  poor  accuracy  can  be  made  by  using  a rated 
steam  engine  or  other  heat  engine  as  the  driving  machine  in 
place  of  a rated  electric  motor,  in  which  case  the  input  to  the 
engine  is  obtained  by  means  of  indicator  cards. 

The  efficiency  of  direct-connected  steam  engines  and  alterna- 
tors, especially  of  steam  turbo-generator  sets,  is  often  given  in 
terms  of  pounds  of  dry  steam  per  kilowatt  of  output,  in  which 
case  the  input  is  determined  from  the  condensed  steam  at  the 
engine  exhaust.  Or  when  the  efficiency  is  given  in  terms  of  the 
ratio  of  power  output  to  input,  the  input  is  measured  by  means 
of  a steam  engine  indicator,  when  this  is  possible.  The  output 
is  always  to  be  measured  in  such  cases  by  carefully  calibrated 
wattmeters.  It  is  of  course  understood  in  the  case  of  a steam 
turbine  set  that  the  mechanical  power  delivered  to  the  dynamo 
is  difficult  to  determine,  but  it  may  be  approximately  computed 
by  determining  the  no-load  losses  of  the  alternator  and  turbine 
rotor  by  driving  the  alternator  as  a synchronous  motor  and 
using  the  data  thus  measured  as  a basis  for  computing  the 
dynamo  input  from  its  output  plus  its  losses. 

In  the  case  of  water  turbines  the  input  to  the  turbine  can  be 
determined  by  the  head  and  rate  of  flow  of  the  water  used. 

The  following  methods  are  especially  adapted  to  shop  testing 
of  the  heating  and  efficiency  of  an  alternator. 

2.  Feeding  Bach  Method  for  measuring  Efficiency.  — It  is 
sometimes  desirable  to  make  a test  with  actual  full  load,  while 
avoiding  the  use  of  a power  dynamometer  or  the  necessity  for 
supplying  the  full  rated  input  from  the  shop  circuits.  With 
this  in  view  a modification  of  Hopkinson's  method  of  testing  * 
may  be  made  (Fig.  434).  Two  equal  alternators  are  rigidly 
coupled  together  in  proper  step  for  parallel  working.  Their 
armatures  are  electrically  connected  together  with  a wattmeter 
and  an  amperemeter  in  the  circuit.  The  field  magnets  being 
properly  excited  by  a separate  exciter,  so  that  one  machine  will 
act  as  a generator  and  the  other  as  motor,  the  system  may  be 

* Jackson’s  Electromagnetism  and  the  Construction  of  Dynamos,  p.  256. 


SYNCHRONOUS  MACHINES 


733 


driven  by  supplying  only  sufficient  power  to  make  up  the  losses 
of  the  two  machines.  Assuming  the  armature  and  stray  losses 
of  the  two  machines  to  be  equal,  and  representing  the  wattmeter 


/T JW 


EXCITER 




RHEOSTAT 


Fig.  434.  — Two  Alternators  connected  up  for  an  Efficiency  or  Heat  Test  by  a Feed- 
ing' Back  Method. 


reading  by  Pe,  the  power  supplied  by  Pm,  and  the  resistance  of 
the  connections  by  i?1;  then  the  efficiency  of  the  generator  is 

= P, , 

if  IfRs  is  the  field  loss  of  the  generator.  If  the  machines  are 
self-exciting,  the  power  in  the  field  circuits  must  be  measured 
and  proper  allowance  made.  The  power  supplied  to  make  up 
the  losses  may  be  measured  by  a rated  direct-current  or  alter- 
nating-current motor  (as  explained  earlier  in  the  article). 

3.  Efficiency  by  Rated  Motor.  — Where  approximate  determi- 
nations of  the  various  losses  of  conversion  and  of  the  commer- 
cial efficiency  are  sufficient,  the  alternator  tested  may  be  driven 
directly  from  a “rated”  direct  or  alternating  current  motor 
as  explained  earlier  in  the  article.  By  means  of  amperemeter 


734 


ALTERNATING  CURRENTS 


and  voltmeter  the  power  supplied  to  the  motor  may  be  deter- 
mined with  the  alternator  operated  under  such  various  condi- 
tions as  may  be  necessar}-'  to  determine  the  losses.  In  each 
case  the  power  supplied  to  the  alternator  is  equal  to  the  power 
absorbed  by  the  rated  motor  multiplied  by  the  efficiency  of  the 
motor  given  in  per  cent  for  its  load  in  the  particular  test.  The 
motor  may  be  rated  by  determining  its  efficiency  by  the  stray 
power  method.  If  the  motor  efficiency  for  various  loads  is 
determined  by  some  exact  method,  the  power  transmitted  to 
the  alternator  may  be  determined  with  considerable  exactness. 
When  it  is  desired  to  carry  the  accuracy  of  this  method  of 
testing  beyond  a fairly  good  approximation,  the  efficiency  at 
various  loads  of  the  rated  motor  should  be  plotted,  so  that  the 
efficiency  at  any  load  may  be  readily  read  off. 

4.  Mordey s Method. — -A  neat  arrangement  for  testing  a 
single  alternator  with  a stationary  armature  by  a method  akin 


Fig.  436.  — Unbalanced,  Divided  Armature  Connection  for  making  Efficiency  or 
Heat  Test  of  Alternator  with  Revolving  Field  Magnet.  Applicable  to  Single  or 
Polyphase  Machine. 


to  Hopkinson's  two-machine  method  was  devised  by  Mordey.* 
It  is  applicable  to  alternators  which  have  stationary  armatures, 
and  where  the  individual  coils  may  be  connected  in  the  desired 
combination.  Such  an  armature  may  be  divded  into  two  parts 
which  are  connected  in  such  a way  as  to  oppose  each  other 
(Fig.  435).  If  one  part  gives  a somewhat  higher  voltage  than 
the  other,  the  first  part  will  operate  as  a generator  and  the 
second  as  a motor  if  the  alternator  is  driven  in  the  usual 

* Testing  and  Working  Alternators,  Join-.  Inst.  E.  E.,  Yol.  22,  p.  116. 


SYNCHRONOUS  MACHINES 


735 


manner.  By  properly  adjusting  the  difference  between  the 
voltages  of  the  two  parts,  the  current  flowing  in  the  machine 
may  be  caused  to  have  any  desired  value.  The  efficiency  of  the 
machine  is  gained  by  measuring  the  power  absorbed  by  the 
machine  operating  as  a self-contained  motor-generator  and 
measuring  its  output  by  a wattmeter,  the  voltage  coil  of  the 
wattmeter  being  connected  across  the  terminals  of  the  motor 
or  generator  coils  and  the  current  coil  being  connected 
directly  in  the  circuit.  The  power  thus  measured  by  the 
wattmeter  is  evidently  approximately  one  half  of  the  total 
energy  of  the  machine;  consequently  the  corrected  efficiency  is 

2 Pe 

V 2 Pe  + Pm  + PR/ 

where  Pm  is  the  power  supplied  to  the  driving  machine  and  Pe 
is  the  power  read  on  the  wattmeter. 

The  difference  between  the  voltages  developed  in  the  parts 
of  the  machine  which  is  necessary  to  cause  the  desired  current 
to  circulate  may  be  caused  by  an  unsymmetrical  division  of  the 
armature  (Fig.  435),  or  by  supplying  a little  additional  vol- 
tage to  the  generator  by  means  of  a transformer.  The  latter 
may  be  supplied 
from  another  alter- 
nator operating  in 
synchronism  with 
the  machine  under 
test  or  it  may  be 
supplied  directly 
from  the  test  ma- 
chine (Fig.  436). 

The  transformer  is 
likely,  however,  to 
introduce  uncertain 
elements  of  loss. 

Figures  435  and  436 
may  be  considered 
to  represent  the  windings  of  a single-phase  alternator  or  one  phase 
of  a polyphase  alternator.  In  the  case  of  polyphase  machines 
each  phase  is  connected  independently  as  in  the  case  of  the  one 
shown  in  the  figures.  In  order  that  the  alternator  may  be 


Fig.  436.  — Divided  Armature  Connection  for  Efficiency 
and  Heat  Tests.  Extra  Voltage  supplied  by  External 
Transformer.  Applicable  to  Single  or  Polyphase  Al- 
ternators with  Revolving  Field  Magnet. 


736 


ALTERNATING  CURRENTS 


tested  for  various  loads,  the  secondary  winding  of  the  trans- 
former shown  in  Fig.  436  should  contain  a number  of  voltage 
taps,  or  a regulating  impedance  may  be  placed  in  series  with 
its  primary  winding.  For  unity  power  factor  this  impedance 
should  consist  of  resistance  and  for  current  lags  of  resistance 
and  reactance.  Instead  of  a transformer  an  autotransformer 
or  compensator  may  be  used.  It  will  be  noted  that  such 
methods  as  this  one  in  which  the  divided  armature  is  used  are 
especially  desirable  for  heat  runs  (that  is,  for  operating  the 
machine  for  a period  under  full  excitation  and  full  load  current 
to  determine  the  rise  of  temperature),  as  only  a small  fraction 
of  the  full  load  power  is  required  to  obtain  full  heating  effects, 
but  they  are  not  so  desirable  for  efficiency  measurements,  since 
dividing  the  armature  alters  the  armature  reactions  and  changes 
the  core  losses. 

5.  Divided  Field  Method  of  measuring  Alternator  Losses  or 
Efficiency.  — As  pointed  out  by  Professor  Ayrton,*  the  arrange- 
ment preceding  may  be  modified  so  as  to  apply  to  alternators 
with  either  rotating  armatures  or  rotating  field  magnets.  In 
this  case  opposite  halves  of  the  field  magnet  are  magnetized  in 
opposite  directions,  and  the  armature  is  short-circuited  through 
an  amperemeter  (Fig.  437).  By  adjusting  the  relative  excita- 
tion of  the  halves  of  the  field  magnet  by  means  of  rheostats  or 
otherwise,  a current  of  any  desired  value  may  be  caused  to  cir- 
culate in  the  armature.  If  the  excitation  is  practically  normal, 
the  measured  losses  of  the  machine  when  any  current  is  flowing 
will  be  approximately  equal  to  those  when  the  machine  is  oper- 
ating normally  on  the  same  current,  provided  it  may  be  assumed 
that  the  losses  in  an  alternator  are  appreciably  equal  when 
driven  as  a generator  and  as  a motor.  This  assumption  is  rea- 
sonable for  most  instances  in  which  the  effects  of  armature  re- 
actions and  leakage  reactance  are  not  large.  The  arrangement 
here  described  is  not  applicable  to  machines  with  armatures 
with  the  halves  wound  in  parallel,  since  under  the  test  condi- 
tions an  amperemeter  connected  between  the  terminals  of  such 
an  armature  would  not  indicate  the  current  circulating  in  the 
armature  coils.  The  plan  does  not  put  full  voltage  on  the 
armature  circuit  and  therefore  does  not  afford  a complete  test, 
besides  altering  the  effect  of  armature  reactions. 

* Jour.  Inst.  E.  E„  Yol.  22,  p.  136. 


SYNCHRONOUS  MACHINES 


737 


Fig.  437.  — Divided  Field  Connection  for  making  Efficiency  and  Heat  Tests.  Appli- 
cable to  Single  and  Polyphase  Alternators. 


6.  Motor -generator  and  Similar  Methods  for  Measuring  Effi- 
ciency, Losses , and  Heating  of  an  Alternator.  — Mordey  also  sug- 
gested the  following  purely  electrical  method  of  testing  alter- 
nators which  have  stationary  armatures.*  The  machine  being 
properly  excited,  one  half  of  the  armature  winding  of  each 
phase  is  connected  as  a generator  to  an  external  load  R (Fig. 
438).  The  other  half  of  the  armature  is  connected  to  another 


Fig.  438.  — Motor-generator  Method  of  obtaining  Efficiency,  Losses,  and  Heating  of 
Single  or  Polyphase  Alternator  having  a Rotating  Field  Magnet. 

* Jour.  Inst.  E.E. , Yol.  22,  p.  122. 

3b 


738 


ALTERNATING  CURRENTS 


alternator  and  driven  as  a synchronous  motor.  The  total  losses 
under  these  conditions  are  evidently  equal  to  the  power  ab- 
sorbed by  the  motor  half  of  the  armature,  minus  the  output  of 
the  generator  half  of  the  armature  ; that  is,  Pm  — Pgl  where  Pm 
and  Pg  are  respectively  the  power  absorbed  by  the  motor  and 
the  output  of  the  generator.  Since  the  output  P a is  the  out- 
put of  only  half  the  armature,  but  the  losses  thus  determined 
are  those  of  the  whole  machine,  the  losses  will  be  approximately 
the  same  when  the  machine  is  run  as  a generator  with  an  output 
of  2 Pg , provided  generator  losses  and  motor  losses  are  the  same. 
The  efficiency  of  the  machine  as  a generator  is  therefore  ap- 
proximately 


The  input  and  output  of  the  machine  under  test  may  be  meas- 
ured by  wattmeters. 

Another  somewhat  similar  plan  is  to  divide  the  stationary 
armature  into  three  divisions  which  are  connected  in  series. 


Fig.  439.  — Synchronous  Motor  Method  of  testing  Rotating  Field  Single-phase 
Alternator.  Armature  divided  into  Three  Parts. 

Two  of  these  are  made  equal  and  are  connected  in  opposition. 
The  third  consists  of  only  a small  portion  of  the  armature. 
The  machine  is  electrically  connected  to  another  alternator  and 
operated  as  a synchronous  motor  under  the  influence  of  the 
small  armature  division  (Fig.  439).  By  a proper  choice  of 
the  number  of  coils  composing  the  small  division  any  desired 
current  may  be  sent  through  the  machine  to  be  tested.  The 
energy  given  to  the  machine  represents  the  losses  in  the  machine 


V = 


2 Pg  + Pm-Pg+l>Rf 


P a + Pm  + I flK{ 


SYNCHRONOUS  MACHINES 


739 


when  operated  as  a generator  and  producing  the  same  current 
with  the  same  excitation.  This  method  is  also  applicable  to 
determining  the  losses  in  machines  with  revolving  armatures 
the  coils  of  which  are  all  connected  in  series.  In  this  case  the 
fields  are  excited  with  the  poles  on  one  half  reversed  (Fig.  440), 
one  half  of  the  field  being  slightly  stronger  than  the  other.  If 
the  armature  is  connected  to  that  of  another  alternator  of  proper 
voltage,  it  will  run  as  a motor.  The  instantaneous  counter- 
voltage  of  the  machine  under  test  depends  upon  the  relative 
strength  of  the  two  halves  of  the  field  magnet,  and  by  adjusting 


Fig.  440.  — Synchronous  Motor  Method  of  testing  Single-phase  Alternator  with  Ro- 
tating Armature.  Field  divided  into  Two  Parts. 


this,  with  due  reference  to  the  impressed  voltage  which  should 
be  of  a relatively  small  value,  the  current  flowing  in  the  arma- 
ture circuit  may  be  given  any  desired  value.  Under  these 
conditions  the  losses  in  the  test  machine  are  equal  to  the  power 
absorbed,  which  may  be  measured  by  a wattmeter.  The  core 
losses  may  be  very  much  altered  from  their  normal  value  by  the 
practice  of  this  method. 

When,  in  either  of  the  cases  mentioned  heretofore,  the  opera- 
tion of  an  alternator  as  a motor  is  predicated,  it  is  assumed 
either  that  the  test  machine  is  brought  to  synchronism  with  the 
alternating  source,  or  that  the  primary  generator  is  started  from 


710 


ALTERNATING  CURRENTS 


a state  of  rest  after  the  circuit  connections  are  made  with  the 
machine  to  be  tested,  in  which  case  the  duly  excited  test  ma- 
chine will  start  and  run  with  the  generator. 

These  methods  of  testing  are  not  only  sometimes  convenient 
in  determining  the  losses  and  the  efficiency  of  an  alternator, 
but  the  tests,  according  to  several  of  the  methods,  are  made 
with  the  consumption  of  comparatively  little  power.  This 
makes  the  methods  satisfactory  for  use  in  shop  tests  for  deter- 
mining the  reliability  in  operation  and  the  heating  limits  of 
machines.  Mordey  suggested  that  the  efficiency  of  an  alternator 
with  stationary  armature  may  be  determined  from  a test  of  one 
armature  coil,  but  this  cannot  serve  as  a satisfactory  shop  test, 
which  requires  a test  of  the  complete  machine. 

7.  Seating  Tests.  — The  heating  of  an  alternator  should  be 
determined  much  as  is  done  in  the  case  of  a transformer.*  The 
machine  should  be  run  at  specified  load  under  normal  condi- 
tions until  the  temperature  has  become  constant.  The  temper- 
atures of  field  and  armature  windings  should  be  determined  by 
both  resistance  measurements  and  thermometers.  The  temper- 
atures of  the  cores,  collector  rings,  bearings,  and  other  parts 
must  be  measured  by  thermometers.  After  a machine  has  run 
until  thermometers  on  the  stationary  parts  have  reached  con- 
stant readings  and  is  then  stopped,  the  thermometers  on  venti- 
lated stationary  parts  will  again  begin  to  rise.  This  is  due  to 
the  heat  from  interior  masses  of  the  constructive  material  tend- 
ing by  conduction  to  equalize  the  temperatures  at  the  surface 
and  interior.  The  temperatures  recorded  for  the  test  should  be 
the  highest  obtained.  The  heating  test  can  sometimes  be  made 
conveniently  by  alternately  running  the  machine  with  the  arma- 
ture short-circuited,  with  the  field  flux  of  such  a value  that  some- 
thing over  full  load  current  flows,  and  then  running  it  with  open 
circuit  armature  and  a field  flux  which  will  give  something  over 
normal  no-load  core  losses  (this  can  be  determined  by  the  power 
input).  The  alternate  periods  of  each  kind  of  operation  should 
be  short,  i.e.  only  a few  minutes.  The  short  circuit  current 
and  its  period  of  flow  should  be  so  adjusted  that  the  product  of 
the  average  I“R  loss  times  the  time  of  the  runs  is  equal  to  the 
kilowatt  hours  which  would  be  expended  by  the  rated  full  load 
current  flowing  constantly  during  the  total  time  of  the  test. 


*Art.  147. 


SYNCHRONOUS  MACHINES 


741 


Likewise  the  iron  loss  and  its  period  of  activity  should  be  so 
adjusted  that  its  average  value  times  the  time  of  the  runs  is 
equal  to  the  kilowatt  hours  which  would  be  expended  by  the 
normal  full  load  excitation  during  the  period  of  the  test. 

8.  Insulation  Tests.  — Dielectric  strength  tests  should  be 
made  as  in  the  case  of  transformers.*  The  voltage  applied 
should  follow  the  accompanying  specification,  which  is  in  general 
applicable  to  all  electrical  machinery.  The  voltage  wave  should 
have  an  approximately  sinusoidal  form.  In  the  case  of  field 
windings,  the  voltage  of  the  exciter  circuit  should  be  used  in 
obtaining  the  test  voltage  from  the  succeeding  table,  except 
where  the  alternator  is  to  be  used  as  a synchronous  motor  and 
is  brought  to  speed  as  an  induction  motor,  when  the  test  voltage 
should  be  5000  volts,  since  high  voltages  are  likely  to  be  induced 
in  the  field  windings  under  such  conditions.  In  testing  the 
dielectric  strength  of  high  voltage  machines  care  should  be  ex- 
ercised, as  in  the  case  of  transformers,  to  bring  the  testing 
voltage  up  very  gradually. 

It  may  be  noted  that  high  voltages  in  a large  machine  may 
cause  appreciable  heating  due  to  the  dielectric  loss.  This 
amounts  to  from  8 to  5 or  more  kilowatts  in  some  of  the  largest 
machines  now  in  service  while  generating  voltages  of  10,000 
volts  and  over. 


TABLE  OF  DIELECTRIC  STRENGTH.  TEST  VOLTAGES  f 


Rated  Terminal  Voltage  ok  Armature 

Rated  Output 

Testing  Voltages 

Under  400  volts 

Under  10  Kw. 

1000 

Under  400  volts 

10  Kw.  and  over 

1500 

Under  800  and  over  400  volts 

Under  10  Kw. 

1500 

Under  800  and  over  400  volts 

10  Kw.  and  over 

2000 

Under  1200  and  over  800  volts 

All 

3500 

Under  2500  and  over  1200  volts 

All 

5000 

Over  2500  volts 

All 

Double  the  normal 

rated  voltage 

9.  Regulation.  — The  regulation  of  a constant  voltage  alter- 
nator may  be  obtained  by  loading  the  machine  to  full  load  at 
normal  speed,  voltage,  and  other  normal  full  load  conditions, 
* Art.  147. 

f Standardization  Rules  of  American  Institute  of  Electrical  Engineers. 


742 


ALTERNATING  CURRENTS 


and  then  removing  the  load  without  change  of  speed.  The 
regulation  is  the  ratio  which  the  difference  between  the  voltage 
at  full  load  and  the  voltage  at  no  load  bears  to  the  voltage  at 
full  load.  Knowledge  of  the  regulation  and  the  efficiency  are 
frequently  desired  for  fractions  of  full  load  and  at  various 
power  factors.  In  the  latter  case  full  load  is  rated  in  kilovolt- 
amperes, and  the  normal  full  load  conditions  of  the  test  must 
include  the  power  factor  concerned.  Except  where  otherwise 
denominated,  however,  testing  is  done  with  loads  of  unity  power 
factor.  When  the  power  factor  is  less  than  unity  with  current 
lagging,  the  excitation  must  be  larger  than  is  required  to 
obtain  equal  voltage  at  unity  power  factor  and  equal  arma- 
ture amperes.  The  losses  are  then  far  greater  at  the  lower 
power  factor,  and  both  the  efficiency  and  regulation  are 
depreciated. 

In  the  combined  regulation  of  direct-connected  units  composed 
of  an  engine  and  constant  voltage  alternator,  the  speed  is  made 
the  normal  value  for  full  load  and  is  supposed  to  be  the  same 
for  no  load.  However,  when  the  load  is  suddenly  removed, 
there  is  a short  fluctuation  of  speed.  The  speed  regulation  is 
then  commonly  taken  as  the  ratio  of  the  greatest  instantaneous 
change  of  the  speed  at  the  instant  the  load  is  removed  to  the 
normal  full  speed.  The  rapidity  with  which  the  load  is  re- 
moved is  apt  to  have  an  influence  upon  this  ratio.  The  instan- 
taneous values  of  the  speed  fluctuation  can  be  obtained  by  a 
sensitive  speed-recording  instrument,  or  the  necessary  data  can 
be  obtained  by  a recording  voltmeter  or  oscillograph  connected 
to  a small  magneto-generator  driven  from  the  engine  shaft,  in 
which  case  the  speed  variation  can  be  calculated  from  the  vari- 
ation in  instantaneous  voltage. 

10.  Wave  Form.  — The  wave  form  of  the  voltage  produced 
by  an  alternator  can  he  obtained  by  means  of  the  oscillograph 
or  otherwise  as  already  explained  at  some  length.* 

As  said  earlier,  the  conditions  surrounding  the  measurements 
of  dielectric  strength,  regulation,  and  other  quantities  during  a 
test  should,  in  America,  be  in  accord  with  the  Standardization 
Rules  of  the  American  Institute  of  Electrical  Engineers,  to 
which  reference  has  already  been  given  earlier  in  this  article, 
unless  other  specifications  are  specially  provided. 


*Art.  157. 


SYNCHRONOUS  MACHINES 


743 


A Power-factor  indicator  is  often  of  service  in  testing  and  also 
for  a permanent  place  on  a switchboard.  This  may  be  built  on 
the  same  principle  as  the  synchroscope  explained  earlier  (Fig. 
418),  or  it  may  be  built  like  a two-phase  induction  motor.  In 
the  latter  case  a current  having  the  phase  of  the  line  current  is 
passed  through  the  windings  of  one  phase,  and  another  having 
the  phase  of  the  voltage  is  passed  through  the  other,  and  the 
torque  of  the  armature  is  then  a function  of  the  phase  angle. 
Revolution  of  the  armature  being  restrained  by  a spring,  the 
degrees  of  rotation  of  the  armature  are  proportional  to  the 
torque  and  therefore  indicate  the  phase  angle.  A pointer  at- 
tached to  the  armature  and  arranged  to  play  over  a scale  com- 
pletes the  essential  parts  of  the  instrument. 

173.  Hunting  of  Synchronous  Motors  and  Other  Synchronous 
Machines.  — All  machines,  such  as  alternating  current  generators, 
synchronous  motors,  and  rotary  converters,  which  depend  upon  a 


Fig.  441.  — Synchronous  Motor  Diagram  showing  the  Effect  of  Hunting. 


series  or  synchronizing  current  to  hold  them  instep,  are  subject  to 
vibratory  irregularities  in  angular  velocity  which  are  given  the 
name  of  Hunting.  In  explanation  of  this  effect  consider  a 
synchronous  motor  of  which  the  voltages  and  current  relations 
are  illustrated  in  the  diagram  of  Fig.  441.  In  this  diagram  the 
impressed  voltage  is  kept  in  a fixed  position  and  is  assumed 


744 


ALTERNATING  CURRENTS 


to  be  of  constant  scalar  value.  The  counter-voltage  E2  is  also 
assumed  to  be  of  constant  scalar  value.  Now  suppose  that  the 
motor  load  has  just  been  reduced  from  a higher  value  to  one 
where  the  normal  relations  of  the  voltages  and  current  are  repre- 
sented by  the  parallelogram  OExRE2  and  the  line  01,  OR  being  the 
resultant  voltage  and  01  the  current.  These  relations  will  not, 
however,  immediately  prevail,  because  in  moving  forward  to 
its  new  space-phase  position  the  rotating  part  of  the  machine  is 
required  to  take  a momentary  speed  somewhat  greater  than 
that  of  synchronism.  The  added  momentum  gained  by  the 
rotor  then  tends  to  drive  it  farther  .ahead  in  space  phase  than 
the  normal  position,  so  that  the  counter-voltage  takes  a position 
such  as  OE2,  which  is  too  far  advanced  for  steady  balance, 
and  the  resultant  voltage  and  current  vectors  at  the  same  time 
move  forward  in  phase  and  fall  off  in  length  as  illustrated  in 
0RX  and  01'.  The  component  of  the  current  in  opposition 
to  the  counter- voltage  also  falls  off  in  length,  and  the  resist- 
ing moment  of  the  load  being  then  greater  than  the  electro- 
magnetic torque  between  rotor  and  stator,  the  extra  momentum 
is  absorbed  in  the  mechanical  load  and  the  speed  is  reduced 
to  normal.  Now  the  resisting  moment  of  the  load  still  being 
greater  than  the  electro-magnetic  torque,  the  rotor  must  fall 
back.  It  then  must  for  an  instant  slow  its  speed  to  below 
that  of  synchronism  and  thus  it  gives  up  to  the  load  some 
of  its  momentum.  When  it  has  reached  the  space-phase  posi- 
tion where  the  counter- voltage  for  the  load  is  normal,  its  speed 
is,  therefore,  less  than  normal,  and  it  drops  back  still  farther, 
as,  for  instance,  so  that  the  counter-voltage  lies  in  OE2" . But 
as  it  falls  back  the  current  increases  and  the  electro-magnetic 
torque  becomes  greater  than  the  resisting  moment  of  the  load. 
In  the  figure  the  limit  to  which  the  rotor  has  fallen  behind 
proper  step  is  supposed  to  have  been  reached  when  the  counter- 
voltage is  OE2"  and  the  current  01" . The  electro-magnetic 
torque  while  the  rotor  is  at  the  retarded  point  is  greater  than 
the  load  torque,  01"  being  above  normal  for  the  load,  and  the 
armature  tends  again  to  gain  speed  in  excess  of  synchronism  and 
step  ahead  to  normal  position.  The  excess  momentum  due  to 
excess  speed  may  again  carry  the  rotor  beyond  the  normal 
point  for  the  load  and  the  whole  process  be  repeated  over  and 
over  again.  If  there  were  no  losses  caused  by  this  vibration,  it 


SYNCHRONOUS  MACHINES 


745 


would  continue  indefinitely  at  full  amplitude.  There  are, 
however,  frictional,  magnetic,  and  electric  losses.  The  frictional 
losses  of  the  rotor  bearings  and  of  windage  absorb  some  of  the 
excess  energy  stored  in  the  rotor  which  produces  the  vibratory 
movement,  just  as  frictional  losses  tend  to  absorb  the  energy 
of  a pendulum  which  has  been  set  swinging.  The  hysteresis 
loss  and  eddy,  or  short-circuit,  currents  in  the  iron  core  and 
field  windings  caused  by  the  changes  of  the  current  value  act 
in  the  same  way.  In  connection  with  the  latter,  loss  is  also 
occasioned  by  the  inductive  action  of  the  armature  magnetic 
flux  as  it  sweeps  through  the  iron  of  the  field  poles  because  of 
the  superimposed  vibratory  motion  of  the  rotor.  If  no  new  dis- 
turbance occurs,  this  energy  loss  ordinarily  causes  the  vibration 
to  rapidly  reduce  to  zero,  and  the  voltages  and  currents  assume 
and  maintain  their  normal  relation.  In  this  case  the  rotating 
part  of  the  machine  has  the  characteristics  of  a torsional  pen- 
dulum, of  which  the  normal  position  gives  counter-voltage  in  the 
position  OR 2 (Fig.  441),  and  which  is  drawn  toward  that  posi- 
tion by  the  electro-magnetic  torque,  but  which  possesses  the 
moment  of  inertia  of  the  rotating  part  of  the  machine  with  re- 
spect to  the  shaft.  The  normal  position  of  rest  of  this  torsional 
pendulum  travels  in  synchronism  with  the  uniform  theoretical 
rotative  speed  of  the  rotor,  and  the  pendulum  vibration  is  super- 
posed on  that  rotative  speed. 

A mechanical  illustration  of  the  conditions  can  easily  be  de- 
vised. Thus,  suppose  a mass  of  given  weight  is  attached  to  the 
lower  end  of  a spiral  spring  and  is  lifted  at  a uniform  rate  by 
a rope  attached  to  the  upper  end  of  the  spring.  Suppose  now 
that  some  of  the  mass  breaks  away  and  thus  causes  the  remain- 
ing mass  to  vibrate  up  and  down.  The  spring  is  first  shortened 
due  to  the  lessened  weight,  but  the  momentum  given  to  the 
mass  causes  it  to  pass  beyond  the  normal  position  that  it  should 
assume.  The  tension  on  the  spring  in  pounds  is  then  less  than 
the  supported  weight,  and  the  mass  falls  back.  When  it 
reaches  normal  position  again,  it  is  rising  with  less  than  the 
speed  of  the  rope  and,  because  of  its  inertia  or  force  of  momen- 
tum, it  falls  below  the  normal  position.  The  spring  is  now 
stretched  so  that  its  tension  in  pounds  is  greater  than  the 
weight  of  the  mass,  which  is  therefore  pulled  back,  and  the 
process  may  be  repeated  over  and  over.  The  molecular  friction 


746 


ALTERNATING  CURRENTS 


of  the  spring  and  the  extra  wind  friction  of  the  mass  due  to  the 
vibratory  motions  dampen  down  the  amplitude  of  the  vibrations 
until  they  become  zero.  The  product  of  the  speed  of  the  rope 
by  the  weight  of  the  mass  is  equivalent  to  the  output  of  the 
motor,  or  its  rotor  speed  times  the  load  torque.  The  weight  is 
equivalent  to  the  rotor  load  torque  ; the  momentum  of  the  mass 
to  the  rotor  momentum  ; and  the  pull  of  the  spring  on  the 
mass  to  the  electro-magnetic  motor  torque. 

The  time  of  vibration  of  the  mass  may  be  expressed 

where  M is  its  weight  and  K the  compressibility  of  the  spring. 
Likewise  the  same  formula  is  approximately  applicable  to  the 
motor,  where  M is  the  moment  of  inertia  of  the  rotor  about  the 
rotating  axis,  and  K represents  the  reciprocal  of  the  rate  of 
change  of  the  electro-magnetic  torque  with  reference  to  the  dis- 
tance of  the  rotor  from  its  normal  instantaneous  position.  It 
is  seen,  therefore,  that  the  time  of  vibration  is  increased  with 
increase  in  weight  and  diameter  of  the  rotor  and  with  decrease 
of  the  rate  of  change  of  electro-magnetic  torque  per  unit  of  dis- 
tance that  the  rotor  is  away  from  its  normal  instantaneous  posi- 
tion. Now  this  latter  factor  depends  upon  the  rate  of  change 
of  the  component  of  the  current  vector  which  is  in  opposition 
to  the  counter-voltage  for  the  various  positions  of  the  rotor 
with  reference  to  its  normal  instantaneous  position.  A studjr 
of  the  diagrams  given  in  Figs.  424,  425,  and  441  will  show  that 
the  angle  -i ]r  between  the  resultant  voltage  and  current  should 
be  small  to  give  the  greatest  proportional  change  of  electro- 
magnetic torque  with  a swing  of  the  rotor  through  a small 
given  angle.  The  armature  impedance  should  evidently  be 
small,  but  also  to  make  -</r  small,  2 m fL  should  be  small  compared 
with  iL  If  R is  very  small  compared  to  2 7r/Z,  the  conditions 
may  permit  the  hunting  to  increase,  eventually  throwing  the 
armature  out  of  step.  This  result  may  also  occur  through  peri- 
odic changes  in  the  angular  speed  of  the  prime  generators  or  of 
the  motor  load,  provided  the  period  of  such  changes  chances  to 
approach  equality  with  the  natural  period  of  vibration  of  the 
rotor  about  its  axis  as  shown  in  the  preceding  formula. 

If  the  field  windings  are  of  low  resistance,  they  act  in  effect 
like  the  short-circuited  secondary  of  a transformer  to  prevent 


SYNCHRONOUS  MACHINES 


747 


Fig.  442.  — Amortisseurs  for  preventing  Hunting. 

The  damping  effect  of  these  windings  due  to  the  currents 
caused  by  the  rotor  vibrations  may  be  understood  when  it  is 
remembered  that  the  armature  magnetic  flux  for  a polyphase 
synchronous  motor  is  normally  stationary  with  reference  to  the 
poles  of  the  field  magnet  in  case  the  machine  and  line  voltages 
are  in  exact  isochronism.  Then  when  relative  movements  or 
vibrations  of  the  armature  occur  with  respect  to  its  normal 


the  sweeping  back  and  forth  of  the  armature  flux;  that  is,  they 
reduce  the  apparent  reactance  of  the  armature  and  increase  its 
stability.  Instead  of  lowering  the  resistance  of  the  field  wind- 
ings, short-circuited  squirrel  cage  conductors  are  sometimes  laid 
in  the  faces  of  the  field  poles,  as  shown  in  Fig.  442.  These  are 
sometimes  called  Amortisseurs  or  Damping  grids.  They  form 
excellent  short-circuited  secondary  paths  for  the  production  of 
currents  by  the  vibrating  armature  flux  and  the  dissipation  of 
the  energy  of  vibration  by  its  conversion  into  heat. 


Amo 


748 


ALTERNATING  CURRENTS 


instantaneous  positions  with  reference  to  the  field  magnet,  the 
magnetic  flux  of  the  armature  currents  acts  upon  the  ainortisseur 
windings  and  other  metal  parts  of  the  field  magnet  exactly  as 
though  their  vibratory  motion  was  the  only  one  that  existed. 
In  accord  with  the  principles  of  electro-magnetic  induction, 
it  is  to  be  observed  that  the  currents  induced  by  this  mo- 
tion of  the  armature  flux  with  respect  to  the  damping  grids 
produce  electro-magnetic  torques  tending  to  destroy  the  mo- 
tion. This  is  similar  to  considering  that  the  mass  and  spring 
used  in  the  earlier  illustration  are  vibrated  while  the  rope  is 
stationary,  Avhich  is  entirely  justifiable,  as  the  vibrations  have 
no  effect  on  the  upward  movement  of  the  rope.  The  effect  of 
the  amortisseur  windings  may  also  be  compared  to  the  damp- 
ing effect  which  will  be  produced  upon  the  weight  by  a 
medium  having  greater  frictional  resistance  than  air,  such  as 
would  be  the  case  if  the  weight  were  immersed  in  water  or  oil. 

As  noted  earlier,  amortisseur  windings  are  of  service  in  start- 
ing polyphase  synchronous  motors  by  means  of  rotating  field 
induction  motor  torque. 

174.  The  Synchronous  Condenser.  — It  has  been  shown  that  a 
synchronous  motor  causes  a quadrature  lagging  current  to  flow 
in  the  line  when  its  field  magnet  is  under-excited,  and  that  it 
causes  a leading  current  to  flow  when  it  is  over-excited.  In 
the  above  statement  normal  excitation  is  taken  as  that  which 
gives  unity  power  factor  to  the  input  of  the  motor,  for  the  load 
under  consideration. 

This  characteristic  of  the  synchronous  motor  makes  it  valu- 
able for  use  as  a power  factor  regulator,  and  when  so  used  it  is 
called  a Synchronous  condenser. 

As  an  illustration  of  this  service,  consider  a long  transmission 
line  loaded  at  the  receiving  end  with  induction  motors  which 
cause,  during  a certain  period  when  their  combined  mechanical 
load  is  equal  to  500  kilowatts,  a lagging  power  factor  of  80  per 
cent.  Then  the  total  kilovolt-amperes  are  625  and  the  quad- 
rature kilovolt-amperes  are  375.  The  kilovolt-ampere  triangle 
for  these  conditions  is  marked  ABC  in  Fig.  443.  As  the 
sides  of  this  triangle,  AB,  BC , and  AC,  respectively,  equal  the 
impressed  voltage  multiplied  by  the  active  current,  by  the  lag- 
ging quadrature  current,  and  by  the  total  current,  they  may  be 
scaled  to  represent  either  currents  or  kilovolt-amperes.  If  it  is 


SYNCHRONOUS  MACHINES 


749 


C D 

Fig.  443.  — Diagram  showing  the  Components  of 
Apparent,  True,  and  Quadrature  Power  in  a 
Line  supplying  an  Inductive  Load ; and  the 
Rating  of  the  Synchronous  Condenser  re- 
quired to  increase  the  Power  Factor  to  Unity. 


desired  to  increase  the  power  factor  of  this  circuit  to  unity,  a 
synchronous  motor  supposed  for  the  present  to  be  without 
losses  may  be  employed  at  the  receiver  end  of  the  line,  which 
is  excited  so  as  to  demand  from  the  circuit  375  leading  quadra- 
ture kilovolt-amperes. 

Since  the  motor’s  input  is 
limited  by  the  heating  of 
the  armature  conductors, 
its  rated  capacity  must 
be  at  least  375  kilovolt- 
amperes. Mechanically 
it  need  not  have  this 
rating,  for  if  no  load  is 
placed  upon  its  shaft,  the 
only  torque  required  is 
that  sufficient  to  over- 
come losses  of  rotation. 

If  the  voltage  of  supply 
is  assumed  to  be  10,000 
volts  and  three  phase,  the  quadrature  current  of  the  synchro- 
nous condenser  is  12.5V3  amperes.  Since  such  a motor  in  fact 
has  some  losses,  a little  true  power  must  be  supplied  to  it,  which 
is  represented  in  the  figure  by  BE  — CD,  so  that  the  actual 

kilovolt-amperes  of  the 
motor  armature  are  rep- 
resented by  CE. 

If  it  is  desired  to  in- 
crease the  power  factor 
from  .80  to  .95,  the  re- 
lations are  as  shown  in 
Fig.  444.  In  this  case 
the  angle  of  lag  must  be 
reduced  from  36°  45'  to 
18°  15'.  To  bring  this 
about,  the  synchronous 
condenser  must  provide 
approximately  211  kilovolt-amperes  in  leading  quadrature, 
which  comes  to  7 v/3  amperes  in  a three-phase  circuit  at  10,000 
volts.  The  power  losses  of  the  condenser  are  assumed  to  equal 
CD,  the  quadrature  kilovolt-amperes  necessary  to  be  provided 


C D 


Fig.  444.  — Diagram  for  determining  the  Rating 
of  an  Unloaded  Synchronous  Condenser  required 
to  increase  the  Power  Factor  of  a Load. 


750 


ALTERNATING  CURRENTS 


by  the  condenser  are  equal  to  DE , and  the  lagging  component  of 
the  kilovolt-amperes  still  in  the  line  is  equal  to  BF.  The 
rated  loading  of  the  condenser  must  be  CE  kilovolt-amperes. 


Fig.  445.  — Diagram  showing  Power  or  Current  Relations  in  a Circuit  containing  a 
Loaded  Synchronous  Condenser,  which  raises  the  Line  Power  Factor  to  Unity. 

When  it  is  desired  to  increase  the  power  factor  and  at  the 
same  time  obtain  mechanical  power  from  the  motor,  the  dia- 
grammatic constructions  are  as  in  Figs.  445  and  446.  In  Fig.  445 
the  synchronous  condenser  is  arranged  to  deliver  185  kilowatts 
of  mechanical  power  and  is  then  supposed  to  have  losses  equal 


Fig.  446.  — Diagram  showing  Power  or  Current  Relations  in  a Circuit  containing  a 
Loaded  Synchronous  Condenser.  The  Line  Power  Factor  is  increased  by  the  Con- 
denser. 

to  15  kilowatts.  Therefore,  to  bring  the  circuit  to  unity  power 
factor,  it  must  receive  from  the  mains  200  = BE  kilowatts  of 
power,  375  = DE  — — BC  of  quadrature  kilovolt-amperes,  and 
a total  of  425  = CE  total  kilovolt-amperes.  In  Fig.  446  the 
synchronous  condenser  is  adjusted  to  only  reduce  the  quadra- 
ture power  by  200  = DE  kilovolt-amperes,  although  it  receives 


SYNCHRONOUS  MACHINES 


751 


the  same  true  power  as  in  Fig.  445.  It  will  be  noted  that  the 
true  power  load  of  the  condenser  itself  has  an  influence  on  the 
angle  of  lag  of  the  line  current,  for  in  Fig.  446  the  reduction  of 
the  lag  angle  would  have  been  less  by  the  angle  FAE  had  the 
motor  run  without  mechanical  load  but  had  still  provided  200 
= DE  quadrature  kilovolt-amperes.  It  will  also  be  noticed  that 
the  total  loading  of  the  synchronous  condenser  of  Fig.  446  is 
only  282  kilovolt-amperes  when  receiving  200  = CD  kilovolt-am- 
peres true  power  and  200  = DE  kilovolt-amperes  quadrature 
power.  It  is  evident  from  the  construction  that  the  smallest 
motor  capacity  is  required  for  a given  arithmetical  sum  of  true 
and  quadrature  volt-amperes  when  the  two  components  are  equal. 

In  general  we  have  the  loading  required  for  a synchronous 
condenser, 

Ft  = V/V  + Pq\  or  Tt  = V7a2  + lq\ 

where  Pt , Pa , and  I\  are  the  total,  active,  and  quadrature  volt- 
amperes  received  by  the  machine,  and  It , Ia , Iq , are  the  total, 
active,  and  quadrature  currents. 

From  the  current  locus  of  the  synchronous  motor,  it  may  be 
seen  that  the  quadrature  kilovolt-amperes  for  no  mechanical  load 
on  such  machines  make  quite  a large  figure  when  the  fields  are 
over-excited,  and  that  these  increase  up  to  a certain  point  as 
the  load  is  increased  (see  current  locus  FGr"H \ Fig.  430).  This 
may  also  be  seen  by  drawing  a number  of  constant  power  cur- 
rent loci  for  a synchronous  motor  as  in  Fig.  431,  then  for  a given 
over-excitation  which  gives  a current  locus  such  as  SPS'  in 
that  figure,  the  quadrature  leading  current  increases  with  the 
load,  thus,  the  quadrature  leading  current  increases  from  OR  at 
no  load  to  OQ  as  the  load  increases  from  that  represented  by 
the  locus  M"  to  the  constant  load  locus  that  would  pass  through 
the  point  P.  For  further  increase  of  mechanical  load  the 
quadrature  current  decreases.  The  line  QP  is  parallel  to  OEv 
and  the  point  P is  the  tangent  point  between  QP  and  SPS'. 
It  is  therefore  impossible  to  exactly  neutralize  the  lagging  cur- 
rent of  induction  motors,  when  the  loads  vary,  by  the  use  of 
loaded  synchronous  motors  with  fixed  excitation.  Thus,  if  the 
synchronous  motor  is  excited  so  as  to  supply  the  right  quad' 
rature  current  when  driving  its  average  mechanical  load  and 
when  the  load  of  the  induction  motors  on  the  line  is  average,  it 


752 


ALTERNATING  CURRENTS 


will  supply  more  or  less  than  enough  if  its  mechanical  load  in- 
creases or  the  load  of  the  induction  motors  decreases,  and  vice 
versa  if  the  synchronous  motor  load  falls  off  or  that  of  the  in- 
duction motors  increases.  However,  by  setting  the  synchronous 
motor  excitation  at  the  average  value  referred  to,  the  actual 
variation  from  the  correct  value,  for  commercial  circuits,  is  not 
likely  to  vary  more  than  a few  per  cent.  When  necessary,  the 
excitation  can  be  changed  to  more  closely  adjust  the  balance. 
This  is  usually  done  by  hand,  but  if  the  need  is  sufficiently  great, 
an  automatic  regulator  can  be  constructed,  using  a phase-indi- 
cating device  in  the  main  circuit  to  operate  relays  for  control- 
ling the  field  rheostat  or  exciter  rheostat  of  the  condenser. 

Synchronous  condensers  are  desirably  placed  on  a part  of  a 
transmission  line  near  a group  of  apparatus,  such  as  induction 
motors,  which  cause  the  flow  of  lagging  current.  The  con- 
denser then  performs  a twofold  service  ; that  is,  it  improves  the 
line  regulation,  and  at  the  same  time  reduces  the  I2R  loss  in 
the  line  for  a given  number  of  kilowatts  transmitted.  Thus, 
suppose  the  line  supplying  the  load  diagrammed  in  Fig.  443  is 
compensated  to  unity  power  factor.  The  line  loss  which  was 
proportional  to  (625)2  becomes  proportional  to  (500)2  or  it  has 
been  reduced  to  close  to  of  its  original  value.  This  saving  in 
power  in  itself  would  in  many  cases  warrant  the  installation 
of  a synchronous  motor  to  float  without  mechanical  load  on 
the  line.  If  it  is  not  desired  to  reduce  the  line  loss,  the  sav- 
ing in  copper  would  be  nearly  | of  that  which  would  be  de- 
manded without  the  condenser,  for  the  instance  mentioned. 
If  the  total  copper  is  large,  the  saving  may  be  very  material, 
especially  if  the  uncorrected  power  factor  is  very  low.  If  the 
synchronous  condenser  replaces  other  motors  by  carrying  their 
mechanical  load  and  has  a sufficient  rating  so  that  it  can  carry 
an  equal  quadrature  load  (Fig.  446),  the  total  kilovolt-amperes 
rating  required  over  that  of  the  motors  replaced  is  in  the  ratio 
of  1 to  V2.  If  the  quadrature  power  to  be  neutralized  is  not 
great,  the  increase  may  be  but  a small  percentage  of  the  power 
of  the  total  apparatus  installed. 

The  line  regulation  may  be  improved  by  the  use  of  the 
synchronous  condenser  from  two  causes,  first,  by  changing 
the  current  flowing  and  bringing  it  into  phase  with  the  voltage, 
and  thus  changing  the  IZ  drop  and  bringing  the  IX  component 


SYNCHRONOUS  MACHINES 


753 


to  a more  favorable  vector  position  ; and  second,  by  the  auto- 
matic tendency  in  the  synchronous  motor  to  maintain  constant 
voltage.  Thus,  referring  to  the  second  cause,  if  the  line 
voltage  drops,  the  motor  will  have  a relatively  higher  over- 
excitation and  thus  draw  additional  leading  current  from  the 
line,  which  will  tend  to  strengthen  the  fields  of  the  generating 
apparatus  and  neutralize  the  effect  of  line  inductive  reactance. 
If  the  line  exceeds  normal  voltage,  the  counter-voltage  of  the 
motor  will  be  less  in  proportion  thereto,  and  the  motor  will 
draw  less  leading  current  from  the  circuit,  thus  leaving  un- 
neutralized inductive  reactance  in  the  line ; or  if  the  rise  of 
line  voltage  is  sufficiently  great,  the  motor  may  draw  lagging 
current  from  the  line,  and  thus  directly  add  to  the  inductively 
reactive  element  of  the  circuit  impedance. 

The  first  of  these  two  effects  is  illustrated  in  Fig.  447,  in 
which  OE  represents  the  voltage  vector  at  the  receiver  end 
of  the  line,  01  the  current  vector  delivered  by  the  line  at  the 
receiver  end,  and  OE0  the  voltage  vector  at  the  generator 


'O 


0 


I 


0 


jr 

| LOAD  OF  UNITY  POWER  FACTOR 


E 


0 


754 


ALTERNATING  CURRENTS 


end  of  the  line.  In  the  upper  diagram  of  this  figure,  the 
current  lags  behind  the  voltage,  but  gives  the  active  component 
OA  which  is  proportional  to  the  delivered  power.  The  vector 
EE'  parallel  to  01  represents  the  IR  drop  of  the  line,  and 
the  vector  E'E0  in  leading  quadrature  with  01  represents 
the  IX  drop  of  the  line.  Hence  EE 0 is  the  IZ  drop  and 
OE0  = OE  4-  EE0  is  the  generator  voltage.  When  the  same 
power  is  delivered  by  the  line  at  the  same  voltage,  but  the 
power  factor  is  brought  to  unity  by  means  of  a synchronous 
condenser,  the  current  takes  the  value  OR  shown  in  the  middle 
diagram  of  the  figure,  which  is  equal  to  OA  of  the  first  diagram. 
In  this  case  EE'  and  E'E0  are  shorter  than  in  the  first  dia- 
gram, in  the  ratio  of  the  lengths  and  are  respectively  in 

parallel  and  quadrature  with  OE.  Under  these  circum- 
stances the  generator  voltage  OE0  does  not  differ  from  the 
receiving  voltage  OE  by  as  many  volts  as  in  the  first  diagram. 
Now,  if  the  quadrature  current  of  a synchronous  condenser 
causes  the  current  to  lead  the  voltage  at  the  receiving  end 
sufficiently  to  bring  the  power  factor  to  the  same  numerical 
value  as  in  the  first  diagram,  the  active  component  remaining 
equal  to  OA , the  current  is  illustrated  by  01"  in  the  bottom 
diagram  in  the  figure  which  has  the  same  length  as  01  in 
the  first  diagram.  The  IZ  drop  is  EE0 , which  has  the  same 
length  as  EE0  in  the  first  diagram,  hut  its  position  now  results 
in  still  less  difference  between  the  numerical  values  of  OE0 
and  OE  than  in  the  second  diagram.  By  still  further  increas- 
ing the  current  lead,  the  generator  voltage  required  to  trans- 
mit the  power  over  the  line  may  become  smaller  than  the 
voltage  maintained  at  the  receiver  end. 

Synchronous  condensers  are  sometimes  useful  in  generating 
stations,  where  they  care  for  the  lagging  current  coming  in  on 
the  line.  Synchronous  motors  of  the  ordinary  type  may  be 
used  as  synchronous  condensers ; or  rotary  converters  * may 
be  so  used,  in  which  case  the  true  power  output  of  the  machine, 
if  any,  is  electrical  instead  of  mechanical. 

175.  Converters.  — Reference  has  already  been  made  to  the 
possibility  of  converting  a direct-current  dynamo  into  a single- 
phase alternator,  and  the  fact  that  a machine  so  constructed,  with 


* Art.  175. 


SYNCHRONOUS  MACHINES 


755 


commutator  and  alternating-current  collector  rings,  may  be  used 
as  a rotary  converter  to  convert  alternating  currents  into  direct 
currents,  or  vice  versa , or  as  a double-current  machine  to  gen- 
erate or  absorb  both  alternating  and  direct  currents  at  the  same 
time.*  Similar  combinations  may  also  be  applied  to  polyphase 
machines.  When  such  a machine  receives  alternating  currents 
through  its  collector  rings,  running  as  a synchronous  motor, 
and  gives  out  direct  currents  through  its  commutator,  it  is 
called  a Converter  or  Rotary  Converter.  When  it  receives  di- 
rect current  through  its  commutator,  running  as  a shunt- wound 
motor,  and  gives  off  alternating  currents  through  its  collector 
rings,  it  is  called  an  Inverted  Converter  or  Inverted  Rotary. 

It  is  possible  in  this  manner  to  make  a quarter-phaser  to 
be  used  with  separate  circuits  out  of  any  direct-current  ma- 
chine with  a closed  circuit  armature  winding,  such  as  the 
Gramme  or  Siemens  winding,  by  arranging  four  collector  rings 
on  the  shaft  and  connecting  them  to  the  armature  windings  at 
points  which  are  90  electrical  degrees  apart.  It  is  also  pos- 
sible to  make  a three-phaser  out  of  a direct-current  machine 
by  arranging  three  collector  rings  on  the  shaft,  and  connecting 
them  to  the  armature  windings,  at  points  120  electrical  degrees 
apart.  Such  machines  may  be  used  as  converters  to  transform 
direct  currents  into  polyphase  currents  or  vice  versa.  Likewise, 
a six-phaser  can  be  constructed  by  using  six  collector  rings  con- 
nected to  the  armature  winding  at  points  60  electrical  degrees 
apart. 

In  Fig.  448  is  shown  a diagram  of  a multipolar  machine  with 
the  connections  of  the  armature  winding  indicated  from  A to  B. 
The  distance  from  A to  B is  360  electrical  degrees,  so  that  the 
taps  to  be  used  for  a single-phase  machine  are  those  numbered 
2 and  5.  The  left-hand  tap  marked  2 is  the  first  connection 
for  the  succeeding  360  electrical  degrees  of  armature  circum- 
ference, and  it  must  also  be  connected  to  ring  2 to  complete  the 
circuit  through  the  windings  of  the  second  180°.  For  a quarter- 
phase  machine  rings  and  taps  2,  1,  5,  and  8 are  used;  for  a tri- 
phase machine  the  taps  and  rings  to  be  used  are  marked  2,  4, 
and  6 ; while  for  a six-phase  machine  they  are  2,  3,  4,  5,  6,  and  7. 

Suppose  now  a commutator  is  attached  to  the  windings  of  the 
machine  as  shown  in  Fig.  448,  with  the  brushes  for  connecting 


* Art.  27. 


756 


ALTERNATING  CURRENTS 


to  direct-current  circuits  placed  upon  the  commutation  points 
as  indicated,  then  the  machine  so  constructed  may  be  used  as  a 
converter.  It  will  run  in  synchronism  when  fed  with  alternat- 
ing currents  through  the  collector  rings,  and  its  speed  therefore 
depends  upon  the  number  of  poles  in  the  field  magnet  and  the 


Fig.  448.  — Diagrammatic  Arrangement  for  showing  Alternating  and  Direct-current 
Connections  for  a Closed  Circuit  Armature. 


frequency  of  the  currents  fed  to  it.  Polyphase  converters  are 
generally  self-starting  from  the  alternating-current  end  by  the 
effect  of  eddy  currents  set  up  in  the  pole  pieces  by  the  rotary 
field  which  exists  in  the  armature  when  polyphase  currents  are 
passed  through  it.  The  starting  torque  may  be  increased,  as  in 


SYNCHRONOUS  MACHINES 


757 


polyphase  synchronous  motors,  by  embedding  short-circuited 
copper  bars,  or  amortisseur  windings,  across  the  pole  faces. 
After  a converter  fed  by  alternating  currents  is  in  synchron- 
ism, its  field  magnet  may  be  magnetized  by  the  direct  current 
produced  by  itself  and  collected  from  its  commutator. 

In  connecting  the  armature  windings  of  converters  to  the 
collector  rings,  the  relative  angles  corresponding  to  the  current 
phases  must  be  carefully  distinguished  as  indicated  above  in 
the  description  of  Fig.  448.  One  complete  revolution  of  an 
armature  in  a two-pole  field  corresponds  to  one  complete  period 
of  the  alternating  current,  and  therefore  860  mechanical  de- 
grees corresponds  to  360  electrical  degrees;  but  in  multipolar 
machines  an  extent  of  rotation  of  the  armature  equal  to  twice 
the  angular  pitch  of  the  poles  corresponds  to  one  complete 
period,  so  that,  in  general,  the  relation  of  electrical  degrees  to 
mechanical  degrees  is  p : 1,  where  p is  the  number  of  pairs  of 
poles  in  the  field  magnet.  Two-pole  converters  evidently  utilize 
the  whole  of  the  armature  winding  with  each  collector  ring  con- 
nected to  a single  point  on  the  winding,  and  the  same  is  true  of 
multipolar  machines  with  two-path  armature  windings.  If  sin- 
gle connections  to  the  collector  rings  should  be  used  in  multi- 
polar machines  with  multiple-path  armature  windings,  a portion 
only  of  the  armature,  corresponding  to  360  electrical  degrees, 
would  be  occupied  in  utilizing  the  alternating  currents ; the 
armature  capacity  would  therefore  be  not  fully  utilized  and  the 
machine  would  be  seriously  unbalanced.  To  fully  utilize  the 
armature  in  this  case,  each  collector  ring  must  be  connected  to 
the  winding  at  as  many  points  as  there  are  pairs  of  poles,  the 
points  being  360  electrical  degrees  apart.  In  Fig.  448  the  taps 
and  windings  are  shown  only  for  360  electrical  degrees.  Evi- 
dently to  complete  the  connections,  the  commutator  and  wind- 
ings should  extend  over  the  entire  360  mechanical  degrees  and 
there  should  be  a total  of  8 p taps  like  those  shown  in  the 
diagram,  similar  taps  in  each  360  electrical  degrees  of  circum- 
ference being  connected  to  the  same  collector  rings.  In  this 
figure  the  collector  rings  for  single,  quarter,  three,  and  six  phases 
are  all  connected  to  one  armature  winding  for  purposes  of  com- 
parison. It  should  be  understood  that  only  the  number  of  rings 
necessary  on  the  system  for  which  the  machine  is  designed  to 
be  used  are  ordinarily  placed  on  the  armature  shaft. 


758 


ALTERNATING  CURRENTS 


In  general,  for  any  number  of  taps,  the  number  of  collector 
rings  equals  the  number  of  phases  and  the  number  of  electrical 


degrees  between  the  taps  of  a phase  is  equal  to 


360 

IT 


where  N is 


the  number  of  phases.  In  this  case  N for  the  single-phase  sys- 
tem must  be  taken  as  equal  to  2,  since  there  are  really  two 
phases  180°  apart  in  that  system,  and  for  the  quarter-phase 
system  N must  be  taken  as  equal  to  4,  as  the  currents  flowing 
in  the  four  wires  differ  in  phase  by  90°,  when  followed  around 
in  order. 

176.  Ratio  of  Transformation  in  Converters.  — Referring  to 
Fig.  448,  it  may  be  observed  that  when  the  machine  is  driven 
as  a synchronous  motor  by  impressing  alternating  voltages  upon 
a set  of  rings,  the  field  magnet,  armature  windings  and  commu- 
tator constitute  a direct-current  generator.  Consider  a theo- 
retical case  first,  where  the  armature  windings  are  without  re- 
sistance and  the  voltage  generated  by  the  revolving  conductors 
is  always  equal  and  opposite  to  an  alternating  voltage  supplied 
from  a single-phase  system  to  the  winding  section  between  two 
rings  through  which  it  is  supplied,  — the  two  rings  considered 
here  being  those  numbered  2 and  5 in  Fig.  448.  Then,  when 
the  armature  is  in  the  position  shown  in  the  figure  with  the 
points  2 and  5 just  under  the  direct-current  brushes,  the  gen- 
erated and  hence  the  impressed  alternating  voltages  are  a max- 
imum and  equal  and  opposite.  This  is  on  the  assumption  that 
there  are  no  armature  reactions  and  that  the  brushes  are  there- 
fore upon  magnetic  neutral  points.  But  if  there  are  enough 
coils  in  the  winding  and  segments  in  the  commutator  to  make 
the  direct-current  voltage  between  the  brushes  of  constant  value, 
the  voltage  generated,  at  the  instant  under  consideration,  is  the 
constant  direct-current  voltage  between  each  pair  of  brushes, 
which  is  equal  to  the  maximum  value  of  the  alternating-cur- 
rent  voltage.  This  latter  may  be  called  em. 

By  the  assumptions  the  voltage  generated  in  a section,  say 
AC,  is  to  be  opposite  to  the  impressed  voltage  in  the  same  sec- 
tion. When  the  middle  of  this  section  is  under  brush  m,  the 
voltage  is  zero  because  that  generated  in  the  two  halves  of  the 
section  neutralize  each  other.  It  is  again  zero  when  the  mid- 
dle of  the  section  is  under  brush  n.  Suppose  that  the  gener- 
ated and  impressed  voltages  in  this  section  are  of  sinusoidal 


SYNCHRONOUS  MACHINES 


759 


wave  shape,  then,  since  the  direct  voltage  is  constant,  the  re- 
lation between  the  direct  and  alternating  voltages  is 

i?2=^  = .707  em, 

V2 

where  E2  is  the  effective  impressed  alternating  voltage.  But 
em  has  been  shown  to  be  equal  to  the  direct  voltage,  which 
may  be  called  E.  Hence  E^  = .707  E. 

By  the  methods  used  in  the  construction  of  Fig.  33  * it  was 
shown  that  when  a wide  coil  moves  in  a magnetic  field  so  that 
a sinusoidal  voltage  is  induced  in  each  conductor,  the  vector 
voltages  of  the  individual  conductors  form  a vector  diagram, 
of  which  the  resultant  is  the  voltage  between  the  coil  termi- 
nals, and  is  of  such  shape  that  the  junction  points  of  the  com- 
ponent voltages  lie  in  the  arc  of  a circle.  The  greater  the 
number  of  conductors,  if  uniformly  distributed,  or  of  slots  if 
they  are  placed  in  slots  in  a given  width  of  winding,  the  nearer 
the  component  vectors  coincide  with  the  circular  arc,  and  the 
wider  the  coil,  the  longer  the  arc,  so  that  with  a coil  covering 
180  electrical  degrees  it  is  a semicircle.  Let  the  semicircle 
OJGrCDKA  of  Fig.  449  represent  the  vector  polygon  of  the 
component  voltages  of  the  coil  section  AC  in  Fig.  448  and 
semicircle  ALFBHMO  represent  the  similar  polygon  for  sec- 
tion CB  which  acts  in  parallel  with  AC  and  is  therefore  in 
series  opposition  with  respect  to  induced  voltages.  It  is 
assumed  in  this  that  the  windings  are  so  thoroughly  distrib- 
uted and  the  component  voltage  vectors  are  so  short  that  the 
vector  polygon  and  the  circumference  of  the  arc  coincide, 
which  for  the  purpose  causes  no  appreciable  error.  The 
diameter  OA  is  then  the  alternating  voltage  between  the 

E 

terminals  of  either  section  and  equals  — -• 

V2 

Suppose  now  the  rings  2,  1,  5,  and  8 only  are  connected  in 
Fig.  448,  which  results  in  a quarter-phase  machine  in  which 
the  alternating-current  coil  sections  are  one  half  as  long  as  a 
single-phase  section  between  2 and  5.  The  voltage  between 
connectors  2 and  1 is  equal  to  vector  OC  of  Fig.  449  in  length 
and  direction ; that  between  1 and  5 is  equal  to  CA ; that  be- 
tween 5 and  8 is  equal  to  AB ; and  that  between  8 and  the 

* Art.  20. 


TGO 


ALTERNATING  CURRENTS 


B 

Fig.  449.- — Vector  Diagram  for  showing  the  Relation  of  Direct  and  Alternating  Vol- 
tages in  a Rotary  Converter. 


left  hand  connection,  2,  is  equal  to  BO.  Hence  the  ratio  of 
voltage  transformation  for  a quarter-phase  converter  is  ex- 
pressed by  the  relation 

E = ^=.5E, 

4 9 

where  E±  is  the  effective  value  of  alternating  quarter-phase 
impressed  voltage  and  E is  the  voltage  between  direct-current 
brushes. 

Likewise,  if  the  three  rings  2,  4,  and  6 of  Fig.  448  are  used, 
the  tri-phase  vector  voltages  are  represented  by  the  lines  OD, 
J)E , and  FO  in  Fig.  449  and  the  relative  direct  and  alternating 
voltages  are  obtained  from  the  expression, 

E~  = ^r  E=.%V2  E, 

3 2V2 

where  E3  is  the  effective  value  of  tri-phase  alternating  voltage. 


SYNCHRONOUS  MACHINES 


7G1 


In  like  manner  the  voltage  relation  in  a six-phase  converter 
is  found  to  be 

E.  = — — . 354  E, 

2V2 

where  E6  is  the  alternating  six-phase  voltage.  For  twelve-phase 
converters  the  relation  is 


En  = — ~^8  E=  .183  E , 

12  2V2 


where  Eu  is  the  alternating  voltage,  and  E as  before  is  the 
direct-current  voltage. 

In  general,  it  may  be  seen,  from  the  construction  of  Fig.  449 
that 


En  = 


E . 7 r 

yl sm  n' 


where  EN  is  the  alternating  voltage  per  phase  for  N phases  and 
E is  the  direct-current  voltage,  since  length 


OA  = E,  = — . 

1 V2 

Continuing  the  assumptions  heretofore  made  and  neglecting 
all  losses,  then,  if  the  impressed  alternating  voltage  and  current 
are  in  the  same  phase,  we  have 


2 EIp=I’  — sin  JVp, 

V2  N 


where  E is  the  direct-current  voltage,  p is  the  number  of  pairs 
of  poles  in  the  field  magnet,  N is  the  number  of  alternating- 
current  phases,  I is  the  direct  current  in  the  windings  of  any 
one  of  the  armature  phase  sections,  and  I’  is  the  component  of 
alternating  current  in  the  same  section.  It  will  be  noted  that 
the  actual  current  flowing  in  a section  is  the  resultant  of  I and 
E . The  two  sides  of  the  equation  represent  respectively  the 
total  direct-current  output  and  alternating-current  input  of  the 
machine.  Hence, 


2V2  I 


N sin  \T 

N 


7G2 


ALTERNATING  CURRENTS 


Using  the  same  subscripts  for  P as  were  used  with  E to 
designate  numbers  of  phases,  we  have 

77  _ -x/2  T.  T 7 _ 4~\/2  j-.  j,  _ j.  j,  _ 2V2  j 
l 2 — V _ 7 , 73  — 7 , 7 4—  7 , 7 6 — ^ , 

3Va  o 


2 V2 

and  I'9  = — — — 

12  12  x .2u9 


Z 


Noting  the  manner  in  which  the  parallel  currents  join  in  the 
leads  of  converters  of  different  numbers  of  phases,  we  have  the 
direct  current  per  lead, 

Ii  = 2I 


for  converters  of  all  phases.  The  alternating  current  per  lead 
in  a single-phase r is, 


-Gt  = 2 J7,  = 2V2 1=  V2 IL. 


In  a tri-phaser,  the  currents  join  in  a lead  at  an  angle  of  120°, 
and 


J'3i=V3Z3= 


4V2 I= 2V2 
3 3 


For  a quarter-phaser,  since  the  currents  from  the  two  adjacent 
coil  sections  join  in  a lead  at  the  phase  angle  of  90°, 

7'u  = V2  J',=  V2  J=  A j 

V2 

In  a six-phaser,  the  junction  phase  angle  being  60°, 


I'6L=I'6=^I=  ~Il. 


3 


And  in  a twelve-phaser,  the  junction  angle  being  30°, 

I'm-  2x.2597>b=^/-^|  A 


In  the  actual  machine,  where  there  are  losses,  an  additional 
alternating  current  must  flow  in  each  section  of  such  a value 
that, 

Pc  = IcE'Np , 

where  Pc  is  the  watts  loss  due  to  the  rotation  of  the  armature, 
Ic  is  the  additional  current  required,  E'  is  the  effective  voltage 


SYNCHRONOUS  MACHINES 


763 


in  an  alternating-current  winding  section,  2V  is  the  number  of 
phases,  and  p is  the  number  of  pairs  of  poles  in  the  field 
magnet.  Also  as  the  current  flowing  in  the  armature  causes 
an  IB  drop,  E in  the  formula  given  above  for  relations  of 
voltages  and  currents  must  be  the  total  direct-current  voltage 
generated.  If  the  impressed  voltage  and  current  are  not  in 
phase,  there  will  be  an  additional  component  of  quadrature 
current,  I' q = T sin  0 — Ia  tan  6 , where  I'  is  the  total  com- 
ponent of  alternating  current,  Ia  is  the  active  component  of 
current  heretofore  dealt  with  alone,  and  6 is  the  angle  between 
the  impressed  voltage  and  current  in  the  section.  The  larger 
current  causes  a greater  IB  drop  in  the  direct-current  voltage, 
and  as  will  be  seen  later  causes  increased  reactions  on  the 
converter  field  magnet. 

177.  Frequency  and  Voltage  Limitations  in  the  Converter.  — 

Direct-current  generators  usually  have  alternating-current  fre- 
quencies in  the  armature  conductors  which  are  between  8 and 
25  cycles  per  second,  while  the  frequencies  of  alternating-current 
systems  are  usually  from  25  to  60  cycles  per  second.  Since  the 
product  of  number  of  magnet  poles  and  speed  is  proportional  to 
the  frequency  of  the  conductor  current  in  the  usual  direct-cur- 
rent machine,  and  the  speed  is  limited  by  the  permissible  circum- 
ferential velocity  of  the  commutator,  which  ordinarily  should 
not  run  at  a velocity  of  over  3000  feet  per  minute,  the  number 
of  magnet  poles  must  in  general  increase  with  the  frequency. 
The  larger  the  number  of  poles,  the  less  is  the  distance  between 
direct-current  brushes  on  a commutator  of  given  diameter  and 
hence  the  less  the  number  of  commutator  bars  of  particular 
width  that  can  be  placed  in  the  commutator  in  the  arc  between 
adjacent  direct-current  brushes.  As  the  voltage  which  a direct- 
current  machine  of  given  design  can  produce  without  excessive 
sparking  at  the  commutator  is  dependent,  among  other  things, 
upon  the  number  of  commutator  bars  between  adjacent  brushes 
of  opposite  polarity,  it  is  apparent  that  the  higher  the  frequency 
of  the  alternating  current,  the  less  is  the  possible  safe  direct-cur- 
rent voltage  on  a converter.  By  careful  designing  it  has  been 
found  possible  to  build  polyphase  converters  that  operate  satis- 
factorily at  a frequency  of  60  periods  per  second  with  direct- 
current  voltages  of  from  500  to  600  volts,  which  is  the  max- 
imum direct-current  voltage  ordinarily  employed,  and  for  lower 


764 


ALTERNATING  CURRENTS 


voltages.  Difficulty  may  be  experienced  in  designing  very 
small  machines  which  can  be  manufactured  at  a reasonable 
cost  for  the  maximum  frequency  and  voltage  named,  on  account 
of  the  inherently  small  commutator. 

The  commercial  converter,  in  its  design,  is  a combination  of 
an  alternator  and  a direct-current  generator  with  a large  com- 
mutator. Figure  450  represents  a typical  60-cycle,  600-volt 
converter  of  American  design. 


Fig.  450.  — A Typical  Three-phase  Converter. 


At  one  end  of  the  shaft  of  a converter  there  is  ordinarily  in- 
stalled a small  mechanical  or  magnetic  device  which  will  cause 
the  armature  to  move  slowly  back  and  forth  lengthwise  of  the 
shaft.  If  this  was  not  done,  the  converter  would  rotate  with- 
out any  material  movement  longitudinally,  and  grooves  would 
be  worn  in  the  commutator  and  bearings.  The  commutation 
of  the  higher  voltage  converters  is  apt  to  be  quite  sensitive, 
and  if  the  machine  is  not  in  good  order  or  there  are  disturb- 
ances in  the  alternating-current  line,  it  may  “flash  over,”  that 
is,  short-circuit  instantaneously  from  brush  to  brush. 

When  a converter  is  to  be  operated  inverted  (Art.  182)  either 
in  regular  service  or  at  starting,  it  is  usual  to  attach  an  auto- 


SYNCHRONOUS  MACHINES 


765 


matic  switch  actuated  by  centrifugal  balls  on  the  end  of  the 
shaft,  for  the  purpose  of  automatically  causing  the  main  switches 
to  open  in  case  the  armature  races  and  goes  to  too  high  a speed. 
This  is  a precaution  of  great  importance,  since  any  defect  arising 
in  the  field  excitation  may  result  in  the  armature  rapidly  accel- 
erating to  a dangerous  speed  unless  it  is  instantly  disconnected 
from  the  circuit.* 

178.  Heating  of  the  Armature  Conductors  in  Rotary  Con- 
verters. — Making  the  same  assumptions  as  in  the  discussion  of 
the  relations  of  voltages  and  currents,!  the  currents  in  the 
direct  and  alternating  circuits  are  as  shown  in  the  formulas 
already  developed,  but  they  flow  in  opposite  relative  directions 
in  the  coils  since  the  alternating  current  produces,  in  connection 
with  the  field  magnet,  the  motor  torque,  while  the  direct  current 
produces  the  effect  of  a counter-torque.  In  Fig.  451  at  the  top 
(a)  is  a diagrammatic  representation  of  part  of  the  field  magnet 
and  armature  of  a quarter-phase  converter  developed.  The  ver- 
tical lines  at  C and  D represent  two  quarter-phase  taps  to  two 
of  the  four  quarter-phase  collector  rings.  For  convenience 
only  three  armature  conductors  or  coils  C,  F , and  D are  shown, 
though  the  armature  is  supposed  to  be  completely  wound  with 
distributed  windings.  Consider  first  the  current  which  flows  in 
the  middle  conductor  F of  the  section  CD.  The  direct  current 
reverses  in  the  conductor  as  it  passes  under  the  brush  B (the 
armature  is  supposed  to  move  to  the  left),  and  again  when  it 
passes  under  brush  A.  The  current  in  the  outside  leads  from 
brushes  A and  B is  2 I,  where  the  direct-current  component  in 
section  CD  is  I.  The  line  NNX,  drawn  with  respect  to  axis 
XX,  Fig.  451  (6),  is  the  rectangular  curve  of  this  component  of 
the  current  in  conductor  F,  where  the  abscissas  are  in  electrical 
degrees  having  the  same  scale  as  in  Fig.  451,  (a).  The  alter- 
nating component  of  the  current  in  conductor  F,  which  is  as- 
sumed to  be  in  exact  opposition  to  the  counter-voltage  of  the 
windings  and  to  be  sinusoidal,  is  shown  by  the  curve  M,  with 
its  maximum  value  occurring  when  F is  under  the  center  of 
pole  piece  X,  and  the  zero  values  of  the  cycle  occurring  when 
F is  under  the  brushes  A and  B.  It  has  been  shown  f that  in 
a quarter-phase  system  the  effective  alternating  current  between 
C and  D of  the  section  is  equal  to  the  current  between  the 
* Art.  182.  t Art.  176. 


766 


ALTERNATING  CURRENTS 


direct-current  brushes  or  I'  = I.  Hence,  the  maximum  value 
of  current  M is  V2  /,  where  I is  equal  to  one-half  the  current 
entering  the  external  circuit  from  brush  B.  Plotting  the  differ- 


Fig.  451.  — Diagrams  for  Representing  the  Current  Distribution  and  Heating  in  the 
Armature  Coils  of  a Converter. 

ence  between  curves  M and  NNF  for  the  half  period  shown  in 
the  figure  gives  curve  PPP.  This  shows  the  resultant  current 
in  conductor  F,  and  it  is  equal  in  value  and  direction  to  current 


SYNCHRONOUS  MACHINES 


767 


iV’iVTV’  just  as  the  conductor  approaches  or  leaves  the  direct-cur- 
rent brushes  but  flows  in  the  opposite  direction  therefrom  when 
the  conductor  is  within  the  arc  from  45°  to  135°.  Squaring  the 
ordinates  of  the  current  curve  PPP,  and  the  curve  Q results, 
the  ordinates  of  which  are  proportional  to  the  heat  produced  in 
conductor  F as  it  passes  from  B to  A,  and  the  area  of  which  is 
proportional  to  the  average  heating  in  F.  The  height  of  line 
VW  shows  on  the  same  scale  the  heating  that  would  be  pro- 
duced in  TP  by  the  component  NJSfN  alone.  The  relative  areas 
of  curves  Q and  VW  for  a half  cycle,  using  the  X-axis  as  the  base 
line,  show  the  relative  heating  of  conductor  F when  used  in  a 
converter  and  in  an  equivalent  direct-current  generator.  The 
arrows  in  Fig.  451  (a)  are  of  some  service  in  obtaining  a phys- 
ical conception  of  the  conditions  — if  the  full  line  arrows  rep- 
resent the  direction  of  the  direct  current  component  of  current 
between  the  brushes,  the  dotted  arrows  represent  the  direction 
of  the  alternating  component  as  the  center  of  the  section  CD 
passes  from  brush  to  brush. 

Consider  now  conductor  D at  the  end  of  section  CD  ; the  rela- 
tions are  set  forth  in  Fig.  451  (c).  The  alternating  voltage 
line  M,  marked  Ml,  is  as  before,  but  conductor  D does  not  re- 
verse its  direct  current  until  45°  after  the  reversal  in  F , so  that 
the  direct  current  curve  for  D is  NXNXNV  Then  from  0°  to  45° 
of  the  trail  of  conductor  F from  brush  B to  brush  A,  the  alter- 
nating current  component  in  D is  in  the  same  direction  as  the 
direct  current.  This  is  shown  by  the  curves  and  Mv  The 
same  condition  is  repeated  when  D starts  its  second  loop  at  180° 
and  at  every  other  half  period.  The  resultant  of  the  direct  and 
alternating  currents,  in  conductor  D from  —45°  to  225°,  is  given 
in  the  curve  S'PlSP1  and  the  heating  curve  is  TQXQXT' Qv  In 
a quarter-phase  converter  the  maximum  resultant  current  in  the 
end  conductors  of  a section  is  equal  to  2 /as  shown  at  S and  *S", 
and  the  heating  is  four  times  as  great  for  the  instant  as  that 
which  would  be  caused  by  current  /,  the  heating  effect  of 
which  is  shown  by  the  height  of  the  line  VW  as  before.  The 
average  heating  in  D is  nearly  as  great  as  though  I flowed  in 
it  constantly.  The  condition  of  conductor  C duplicates  that 
of  D. 

Conductors  lying  between  F and  D or  F and  C in  the  section 
are  subjected  to  heating  effects,  on  account  of  the  PZR  loss, 


768 


ALTERNATING  CURRENTS 


which  are  intermediate  between  those  of  the  edge  conductors 
C and  D and  the  middle  conductor  F. 

By  making  the  width  CD  correspond  with  the  angle  covered 
by  a section  between  rings  for  any  other  number  of  phases,  that 
is,  for  tri-phases  120°,  six-phases  60°,  twelve-phases  30°,  or 
single-phase  180°,  and  using  the  currents  determined  in  Art. 
176,  the  relative  heating  effect  in  the  conductors  maybe  deter- 
mined by  the  foregoing  process  for  any  converter.  By  thus 
obtaining  and  adding  the  heat  loss  for  all  the  conductors  of 
a section,  the  average  heat  loss  of  the  section  is  determined. 
From  this  may  also  be  obtained  what  may  be  called  the 
apparent  resistance.  The  sum  of  the  areas  of  the  curves  of 
instantaneous  resultant  current  squared  for  the  conductors, 
which  have  been  referred  to  as  heating  curves,  made  to  proper 
scale  and  multiplied  by  the  electrical  resistance  of  the  section, 
gives  the  energy  expended  for  heat  losses  in  the  armature 
section  per  half  cycle  of  the  alternating  current.  Dividing 

this  by  7 r gives  the  average  power,  P',  which  goes  into  the  heat 

pi 

losses.  Then  R' A = , where  I is  one  half  the  current  from  a 

P 

direct-current  brush,  and  R'A  is  the  apparent  resistance  of  the 
alternating-current  section.  The  apparent  resistance  of  a direct- 
ly 

current  armature  circuit,  R"A,  is  manifestly  R'  A multiplied  by  — , 

which  is  the  ratio  of  the  winding  arc  between  the  direct-cur- 
rent brushes  to  the  arc  between  taps  at  the  ends  of  an  alterna- 
ting-current phase  section.  Dividing  R" A by  2 p,  the  number 
of  direct  current  paths  through  the  armature  winding,  gives  the 
apparent  resistance  of  the  entire  armature,  or 

NP' 


7?" 

7?  — 7 A — 
— 0 — 
2 p 


4 pi 2 


In  general,  the  heating  under  the  conditions  can  be  found  in 


this  way : 


I'  = 


2V27  * 


where  I'  and  I are  the  alternating  and  direct  currents  respec- 
tively in  an  armature  section,  and  N is  the  number  of  phases. 

* Art.  176. 


SYNCHRONOUS  MACHINES 


769 


This  formula  was  derived  under  the  assumption  that  the  rota- 
tive losses  were  zero  and  the  angle  of  lag  between  the  alternat- 
ing voltage  and  current  was  zero.  If,  however,  c per  cent  of 
the  total  current  component  entering  the  conductor  is  used  in 
causing  rotation,  the  left-hand  side  of  the  equation  must  be 


increased  by  the  ratio 


and  if  the  power  factor  is  not  unity, 


it  must  be  further  increased  by  the  factor  — — , when  0 is  the 

cos  6 

angle  of  lag.  Hence  the  alternating  current  becomes  for  such 
a general  condition 


I'  — 


2V2  I 

N sin  r^~ 
N 


1 

(1  — c)  cos  6 


From  the  curves  in  Fig.  451  it  is  seen  that  the  component  of 
alternating  current  at  any  instant  for  any  conductor  between 
commutator  bars  at  a distance  8 from  the  middle  conductor  F, 
when  the  angle  of  lag,  6 , is  zero,  is 

i'c  = V2  I'  sin  («  — 6), 

where  a is  the  angular  advance  of  F from  0°.  When  the  lag  is 
not  zero  this  becomes 

i'c  = V2  I'  sin  («  — 8 — O'). 

The  component  of  direct  current  at  any  instant  is  /,  where  I 
is  one  half  the  current  flowing  through  a direct-current  brush 
to  the  external  circuit.  Therefore  the  instantaneous  resultant 
current  in  the  conductor  is 

iR  = I — V2  /'  sin  («  — 8 — O') ; 


and  substituting  the  value  of  I'  in  terms  of  I, 

41  1 


ip  — T — 


N sin 


7 T (1  — C)  COS  0 

~N 


sin  («  — 8 — d). 


Placing ^ • — j — = A , squaring,  and  integrating 

tit  ■ 7 r (1  — c)  cos  0 

Jy  sin  — v ’ 

JSf 


this  function  of  iR  with  respect  to  a between  the  limits  tt  and 


770 


ALTERNATING  CURRENTS 


0,  and  dividing  by  it  to  obtain  the  effective  current,*  we  obtain 
Ij ? = — 2 V = f f"[l  - A sin  («  - 3 - 0)] 2 d«, 

7T  ^0  7T  ♦'0 


or 


i*2  = /2 


^ d2  _ 4 4 cos  (3  + 9 ) 

nr 


When  9 — 0,  this  is  evidently  minimum  for  3 = 0 or  the  middle 
point  of  the  armature  section,  and  maximum  for  the  edges  of  the 
section  where  3 is  a maximum.  The  heating  of  the  conductor  is 


P'  = RJ2 


A2  j \ _ ^4  A cos  (3  + 6 ) 


where  Rc  is  the  electrical  resistance  of  a single  conductor  of  the 
section.  To  obtain  the  mean  heating  in  the  conductors  of 
the  armature  section  assume  that  the  single  conductors  are  of 
negligible  width,  and  then,  by  integrating  the  right-hand  term  in 
the  brackets  of  the  equation  with  respect  to  3 between  the  limits 

and  — — and  dividing  by  the  width  of  the  section,  2-^,  the 
mean  heat  is  found  to  be 


Pr.  = RJ2 


Kf+i)-^4CJcof+>] 


and 


\AN 


• sin  — cos  6 ■ 
N 


Substituting  the  value  represented  by  A gives 

8 1 


P,  = RJ2 


7V7-2  ' 2 7r  (1  — e)2COS20 

iv2  sim  — v J 
N 


+ i-SS 


1 ■ 

(1-c)  . 


When  the  rotation  losses  are  considered  zero  and  the  power 
factor  is  unity,  this  becomes 


Pr  = RJ‘ 


W2  sin2  J 

N 


+ 1 


16“ 

Trl 


Since  Rc  is  the  true  resistance  of  the  conductor,  RJ 2 is  the 
heating  per  conductor  that  would  occur  if  the  machine  were  run 
as  a direct-current  dynamo  with  an  output  per  brush  of  2 Z and 
as  Pc  is  the  mean  heating  per  conductor  when  the  machine  gives 


* Art.  5. 


SYNCHRONOUS  MACHINES 


771 


out  2 1 amperes  per  direct-current  brush  when  running  as  a con- 
verter, the  expression  surrounded  by  brackets  in  the  formulas 
for  Pc  is  equal  to  the  ratio  of  the  power  lost  in  heat  in  the  two 
cases.  The  rated  capacity  of  a machine  is  inversely  propor- 
tional to  the  square  root  of  its  temperature  rise,  and  therefore 
the  expression  within  the  brackets  may  be  considered  as  express- 
ing the  square  of  the  ratio  of  the  capacity  of  the  machine  when 
running  as  a generator  to  its  capacity  when  running  as  a con- 
verter subjected  to  sinusoidal  voltages.  It  will  be  noted  that 
through  much  the  same  method  of  reasoning  it  is  possible  to 
find  the  ratio  of  the  apparent  to  the  true  resistance  of  the 
converter  armature. 

From  the  graphical  constructions  and  the  last  formula  it 
may  be  observed  that  the  capacity  of  a converter  operated  at 
unity  power  factor,  limited  by  the  rise  of  temperature  of  the 
armature,  when  compared  with  the  capacity  of  the  same 
machine  used  as  a direct-current  generator,  is  as  follows  : single- 
phase .85  to  1 ; quarter-phase  1.63  to  1 ; tri-phase  1.34  to  1 ; 
six-phase  1.95  to  1 ; and  12-phase  2.21  to  1.  Also,  comparing 
in  the  same  way,  the  apparent  resistances  are  respectively  1.38 
to  1 ; .38  to  1 ; .56  to  1 ; .26  to  1 ; and  .21  to  1. 

If  the  small  component  of  active  current  required  to  rotate 
the  armature  is  included  in  the  calculations,  the  resistance 
ratios  will  be  slightly  increased  over  those  given.  Also,  if  a 
quadrature  current  flows  in  the  armature,  the  increased  current 
will  cause  a greater  I2R  loss  in  itself  and  will  cause  a further 
loss  by  reason  of  the  shifting  of  the  phase  position  of  the 
current.  To  solve  the  problem  under  those  conditions  the 
values  of  c and  6 must  be  substituted  in  the  general  formula 
for  Pc,  or  if  in  Fig.  451  a quadrature  current  flowed,  it  would 
have  to  be  combined  with  components  JV  and  M and  by  reason 
of  its  position  would  cause  greater  peaks  of  current  and  hence 
greater  heating  ; and  to  obtain  the  heating  effect  of  the  total 
current  of  a converter  in  which  the  current  is  out  of  phase 
with  the  generated  counter-voltage  it  is  necessary  to  add  fo  M 
an  active  component  equal  to  that  necessary  to  supply  the 
rotation  losses,  and  a second  component  equal  to  the  quadrature 
current.  The  processes  may  then  be  carried  through  as 
before.  If  the  impressed  and  counter  voltages  are  not  both 
sinusoidal,  the  heating  is  modified  from  that  computed. 


772 


ALTERNATING  CURRENTS 


It  will  be  noted  both  from  the  formula  for  Pc  and  Fig.  451 
that  the  width  of  the  alternating-current  coil  section  directly 
affects  the  capacity  of  the  machine.  It  is  evident  then  that 
the  type  of  the  winding  may  materially  influence  the  converter 
capacity.  The  figure  also  shows  that  the  heating  is  not  uni- 
formly distributed  amongst  the  armature  conductors,  but  that 
part  of  each  section  is  more  heated  than  the  remainder. 

179.  Armature  Reactions  of  a Converter.  — A consideration  of 
the  basis  of  the  construction  of  Fig.  451  will  show  that  the  mag- 
netic reactions  of  the  armature  upon  the  field  magnet  in  a con- 
verter are  small  when  no  quadrature  current  flows.  Thus,  in 
the  figure,  which  is  for  a quarter-phase  machine,  the  effective 
alternating  current  component  equals  and  is  opposite  to  the 
direct  current  component  in  any  section,  so  that  there  can  be  no 
resultant  reactive  effect  when  integrated  through  the  duration 
of  a cycle.  But  it  is  noted  that  there  is  none  the  less  a result- 
ant current  flowing  of  irregular  double  frequency  wave  shape 
and  irregularly  disposed  in  the  individual  coils.  This  varying 
current  causes  an  alternate  skewing  of  the  field  magnetic  flux 
forward  and  back  as  the  directions  of  the  current  alternately 
predominate  in  the  coils.  In  a single-phase  machine,  having 
alternating  current  coil  sections  180  electrical  degrees  wide,  the 
heavy  current  that  flows  in  a small  part  of  an  alternating  current 
section  of  the  armature  when  the  alternating  currents  in  the 
coils  near  the  edge  of  the  section  are  in  the  same  direction  as 
the  direct  current,  makes  the  reaction  of  double  frequency 
especially  heavy.  By  reason  of  their  changing  the  reluctance 
of  the  field  magnetic  circuit  these  reactions  may  set  up  appreci- 
able double  harmonic  vibrations  in  the  current  in  the  direct- 
current  leads.  As  the  skewing  is  first  one  way  and  then  the 
other  way,  the  brushes  cannot  be  shifted  to  the  point  of  field 
strength  which  gives  the  least  sparking,  as  in  direct-current 
machines.  However,  even  single-phase  converters  are  built 
which  give  excellent  satisfaction,  while  the  smaller  amount  of 
the  resultant  current  variations  in  polyphase  converters  make 
the  effects  mostly  negligible.  Further,  induced  currents  in  the 
pole  faces  of  the  field  magnet  or  in  the  amortisseur  windings 
cut  down  materially  the  variations  in  field  flux. 

A converter  being  equivalent  to  a synchronous  motor,  in 
its  effect  on  the  alternating  current  supply  circuit,  when  the 


SYNCHRONOUS  MACHINES 


“"5V 
I i U 


excitation  is  raised  beyond  the  point  giving  unity  power  factor, 
a leading  quadrature  current  is  drawn  from  the  line  as  explained 
in  Art.  170.  This  is  in  such  a position,  as  may  be  seen  by 
studying  the  relations  shown  in  Fig.  451,  that  it  tends  to 
weaken  the  field  magnet.  Indeed,  the  amount  of  the  quadrature 
current  that  flows  in  any  synchronous  machine  is  such  that 
the  field  magnet  will  have  a strength  which  will  enable  the 
resultant  between  the  impressed  and  counter  voltages  to  just 
drive  the  armature  current  through  the  armature  impedance. 
Hence,  changes  in  converter  excitation  make  only  small  changes 
in  the  direct-current  voltage,  when  the  impressed  alternating 
voltage  is  constant. 

When  a lagging  current  is  drawn  from  the  line  due  to  under 
excitation  of  the  converter  field  magnet,  the  reactions  strengthen 
the  field  magnet.  In  polyphasers,  the  reaction  due  to  the  quad- 
rature component  of  current  is  caused  by  flux  from  the  armature 
which  is  stationary  with  respect  to  the  field  magnet,  i.e.  the  com- 
bined ampere-turns  of  the  several  phases  cause  the  armature  flux 
to  revolve  with  respect  to  the  windings  against  the  direction 
of  the  revolution  of  the  armature  and  at  equal  speed,  thus 
causing  this  flux  to  maintain  a fixed  angular  position  with 
reference  to  the  poles  of  the  field  magnet.* 

180.  Voltage  Control  of  Converters.  Split-pole  Converters.  — 
From  the  discussion  of  armature  reactions  in  the  previous  article 
it  is  evident  that  the  direct-current  voltage  is  approximately 
proportional  to  the  impressed  alternating  current  voltage  in  any 
particular  converter.  The  direct-current  voltage  may  then  be 
controlled  by  any  method  whereby  the  impressed  voltage  is 
controlled,  such  as  by  cutting  out  or  in  active  turns  on  the 
transformers  feeding  a converter,  or  by  using  autotransformers 
in  which  the  secondary  voltage  may  be  varied,  or  using  induc- 
tion regulators  f in  the  leads. 

By  placing  reactive  coils  in  series  with  the  converter  leads, 
or  when  the  converter  is  used  on  lines  where  there  is  other  in- 
ductance, the  impressed  voltage  may  be  varied  somewhat  by 
the  same  processes  as  where  the  synchronous  condenser  is  em- 
ployed.^; An  illustration  of  this  method  is  given  in  Fig.  447. 
Or  this  action  may  be  made  automatic  by  winding  a few  series 
compounding  turns  from  the  direct-current  leads  of  the  con- 

f Arts.  159,  160.  J Art.  174. 


* Art.  152. 


774 


ALTERNATING  CURRENTS 


verter  on  tlie  field  spools,  and  inserting  reactances  of  proper 
values  in  the  alternator  leads.  A special  alternator  may  be 
connected  to  the  converter  shaft,  with  its  own  field  magnet 
having  the  same  number  of  poles  as  the  converter,  and  with 
its  armature  coils  of  each  phase  connected  in  series  with 
those  on  the  converter  armature,  thus  composing  a Synchro- 
nous regulator.  Such  a machine  is  shown  in  Fig.  452. 
The  synchronous  regulator  is  shown  on  the  right-hand  side  of 


Fig.  452.  — Synchronous  Regulator  attached  to  a Converter. 

the  figure.  By  varying  the  field  strength  of  the  regulator,  the 
voltage  delivered  to  the  converter  is  varied. 

As  the  direct-current  voltage  depends  on  the  maximum  value 
and  wave  form  of  the  alternating  counter-voltage,  anjr  arrange- 
ment whereby  the  wave  shape  of  the  counter-voltage  can  be 
varied  may  be  used  to  vary  the  direct-current  voltage.  Thus, 
if  each  field  pole  is  split  parallel  to  the  shaft  into  three  narrow 
poles,  all  having  separate  windings,  the  magnetic  field  distribu- 
tion may  be  made  peaked  by  strengthening  the  middle  pole  part 
relatively  to  the  others,  or  may  be  flattened  by  weakening  the 


SYNCHRONOUS  MACHINES 


775 


center  part.  If  the  outside  pole  parts  are  strengthened  when 
the  middle  part  is  weakened,  or  vice  versa , the  direct-current 
converter  voltage  may  be  varied  several  per  cent.  When  such 
an  arrangement  is  made,  the  machine  is  called  a split-pole  con- 
verter. 

As  in  any  direct-current  machine,  the  direct-current  voltage 
of  a converter  varies  with  the  shifting  of  the  brushes  with  refer- 
ence to  the  field  poles.  In  order  to  vary  the  voltage  conven- 
iently by  this  means,  some  machines  are  built  with  extra  poles 
whereby  the  magnetic  flux  instead  of  the  brushes  can  be  shifted. 
Figure  453  is  a diagram  of  such  an  arrangement.  The  auxil- 
iary I may  be  strengthened  when  the  poles  NS  are  weakened 
or  vice  versa,  thus  shifting  the  center  of  density  of  magnetic 


Fig.  453. — Flux  shifting  Auxiliary  Poles  on  a Converter. 


flux.  The  auxiliary  pole  piece  I is  usually  somewhat  nearer 
what  is  intended  to  be  the  trailing  pole  tip,  thus  leaving  a fairly 
wide  space  for  commutation,  and  it  is  not  an  interpole  for  com- 
mutation. 

181.  Some  Features  of  Converter  Operation. — A converter 
may  be  started  as  a direct-current  motor  from  the  direct-cur- 
rent side,  or  it  may  be  brought  to  synchronism  as  a rotary  field 
induction  motor,  or  again  it  may  be  brought  to  speed  by  some 
external  mechanical  starting  device.  Except  for  the  first-named 
method,  a converter  is  started  like  a synchronous  motor,  and  is 
paralleled  in  the  same  way.  Figure  454  shows  the  connections 
of  two  six-phase  machines  arranged  to  start  as  rotary  field  in- 
duction motors.  A switch  for  breaking  up  the  length  of  the 
shunt  field  windings  is  shown  at  the  right  hand  of  each  con- 
verter, for  the  purpose  of  preventing  excessive  voltages  being 


776 


ALTERNATING  CURRENTS 


induced  in  them  during  the  starting  process  ; and  a switch  for 
short-circuiting  the  series  or  compound  field  winding  is  at  the 
left  hand.  The  direct-current  switchboard  panels  shown  in  the 


THREE  PHASE 
INCOMING  LINE 


Fig.  454.  — Diagram  of  Connections  for  Six-phase  Converters,  including  the  High 
Voltage  Supply  Line,  the  Transformers  with  Starting  Taps,  and  the  Direct-current 
Switchboard  Panels  arranged  for  Electric  Railway  Circuits. 


SYNCHRONOUS  MACHINES 


777 


figure  are  arranged  for  railroad  work  at  from  500  to  600  volts. 
The  neutral  point  of  the  polyphase  circuit  is  shown  ungrounded. 
Keeping  in  mind  this  figure,  the  following  instructions,  taken 
from  those  issued  by  one  of  the  large  manufacturers  of  electrical 
machinery,  give  an  insight  into  the  methods  of  operating  this 
kind  of  machines  when  arranged  for  self-starting  by  the  alter- 
nating current  : 

Close  the  oil  circuit  breaker  on  the  high  voltage  side  of  the 
transformer.  Insert  the  voltmeter  plug  for  the  direct-current 
voltmeter.  Close  the  double-throw  starting  switch  to  the 
starting  position.  The  rotary  converter  should  come  up  to 
synchronous  speed  in  about  30  seconds,  and  lock  into  step,  in- 
dicating this  condition  by  a steady  current  on  the  alternating- 
current  side  of  the  converter,  and  a continuous  deflection  of  the 
direct-current  voltmeter.  When  the  direct-current  voltmeter 
indicates  the  correct  polarity,  close  the  field  break-up  switch 
so  as  to  have  the  field  rheostat  in  the  field  circuit.  When  the 
direct-current  voltmeter  indicates  reversal  of  polarity,  reverse 
the  field  break-up  switch,  thus  reversing  the  shunt  field  and 
connecting  it  directly  across  the  armature.  The  voltmeter 
indicator  will  swing  back  towards  zero.  When  it  reaches  zero, 
throw  the  field  break-up  switch  to  the  lower  position.  If  the 
voltage  now  comes  up  with  the  right  polarity,  proceed  as 
directed  above.  If,  however,  the  converter  fails  to  slip  a 
pole,  and  the  voltage  again  comes  up  with  reversed  polarity, 
it  is  evident  that  the  field  induced  by  the  alternating  currents 
in  the  armature  is  too  strong.  The  starting  switch  should 
then  be  opened  for  a moment,  thus  permitting  the  converter 
to  slow  down  somewhat.  The  starting  switch  should  then 
be  closed  again  in  the  starting  position.  When  the  machine 
is  up  to  synchronous  speed,  and  the  direct-current  voltmeter 
shows  correct  polarity,  throw  the  starting  switch  to  the  run- 
ning position.  Adjust  the  direct-current  voltage  to  the  proper 
value.  Close  the  series  shunt  switch  (when  there  is  a series 
shunt  circuit).  Close  the  direct-current  equalizer  and  negative 
switches  ; then  close  the  direct-current  circuit  breaker  and  the 
positive  switch. 

The  following  instructions  for  connecting  up  rotary  converters 
for  the  first  time  after  they  have  been  installed  come  from  the 
same  source  : 


778 


ALTERNATING  CURRENTS 


“ Connect  the  alternating  current  leads  from  the  same  bus 
bars  to  similar  switches  and  collector  rings  as  compared  with 
the  converters  already  installed.  Place  the  direct-current  brush 
holder  in  the  same  position  with  respect  to  the  field  poles  as 
that  on  the  other  converters,  and  run  the  positive,  negative,  and 
equalizer  leads  through  their  respective  switches  to  the  positive, 
negative,  and  equalizer  bus  bars  of  the  other  converters.  See 
that  the  field  wires  are  brought  out  to  the  corresponding  ter- 
minals of  the  other  converters  and  connected  in  the  same  way. 
Be  sure  that  the  voltmeter  lead  from  the  positive  terminal  goes 
to  the  positive  voltmeter  bus,  and  the  negative  to  the  negative 
bus.  After  carefully  checking  all  wiring  and  seeing  that  all  is 
clear,  start  the  converter  by  closing  the  alternating-current 
switch  in  the  starting  position.  If  the  armature  revolves  in  the 
wrong  direction,  shut  down  and  change  the  alternating-current 
cables  to  the  converter.  If  the  converter  is  two-phase,  reverse 
the  two  leads  of  either  one  of  the  phases,  if  it  is  three-phase,  re- 
verse any  two  leads.  After  the  converter  has  come  to  correct 
speed,  proceed  as  above  for  self-starting  rotaries.  In  first  start- 
ing a converter  it  is  necessary  to  synchronize  all  phases.  This 
can  be  done  in  a two-phase  converter  by  placing  a bank  of  lamps 
equal  to  the  voltage  of  the  converter  across  the  jaw  and  blade  of 
the  switch  on  each  side  of  each  phase.  If  the  converter  is  three- 
phase,  place  a bank  across  each  of  the  three  switches.  Now 
synchronize  as  before,  noticing  the  lamps.  If  they  all  indicate 
the  proper  phase  relation  at  the  same  time,  the  phases  are  wired 
correctly  and  the  switch  may  be  closed  at  the  proper  time.  If, 
however,  one  set  of  lamps  is  bright,  while  the  other  is  dark,  it 
will  be  necessary  to  change  the  main  alternating-current  leads, 
either  at  the  converter  terminals  or  at  the  point  of  the  switch 
running  directly  to  these  terminals.  If  the  converter  is  two- 
phase,  reverse  the  leads  of  either  phase  ; and  if  it  is  three-phase, 
interchange  any  two  leads.  After  this  change  is  made,  check  it 
by  means  of  the  lamps  when  starting  again.  After  synchroniz- 
ing the  converter,  put  load  on  the  direct-current  circuit  and  test 
the  series  field  coil  by  short-circuiting  it  and  noticing  the  direct- 
current  voltmeter.  If  this  is  in  opposition  to  the  shunt  field,  as 
shown  by  the  voltage  increasing  with  the  series  coil  short-cir- 
cuited, reverse  the  lead  connections  to  these  coils.  The  con- 
verter should  now  be  ready  for  paralleling  on  both  sides." 


SYNCHRONOUS  MACHINES 


779 


The  windings  of  a converter  are  of  the  same  general  type  as 
the  windings  on  direct-current  machines  with  reentrant  arma- 
ture coils.  As  there  are  likely  to  be  many  sets  of  direct-current 
brushes  on  account  of  the  multiplicity  of  poles  on  high  fre- 
quency converters,  there  are  many  direct-current  paths  through 
the  armature  winding.  An}r  inequality  of  the  resistances  in 
Taps  to  Collector  Rings 


Equalizer  Connection 

Fig.  455.  — Equalizer  Connections  between  Sections  of  the  Armature  Windings  of 

Converters. 


these  paths  or  in  the  strengths  of  the  individual  magnet  poles 
is  apt  to  unbalance  the  currents  in  the  paths  and  cause  undue 
heating.  It  is  thus  necessary  to  use  equalizer  connections 
between  points  of  equal  potential  in  the  windings,  as  shown  in 
Fig.  455. 

The  direct-current  characteristics  of  a converter  are  very 
much  like  those  of  a direct-current  generator.  Figure  456  shows 
typical  curves  of  regulation,  commercial  efficiency,  and  losses 
for  a machine  of  this  type.  Since  there  is  little  armature 


780 


ALTERNATING  CURRENTS 


reaction,  the  field  and  all  other  losses  except  the  armature  cop- 
per loss  may  be  considered  constant. 

The  efficiency  of  the  converter  is 

IE 

v Te+p  + pr' 

where  I and  E are  the  direct  current  and  voltage,  P is  the  fixed 
power  loss,  including  the  field  and  rotation  losses,  and  R is  the 
apparent  resistance  of  the  armature.  The  efficiency  may  be 
measured  by  placing  wattmeter’s  in  the  alternating  and  direct 
current  leads  respectively,  and  loading  the  machine  to  the 


amount  and  at  the  power  factor  desired.  Or  the  losses  ma}r 
be  determined  by  running  the  motor  unloaded  at  the  excitation 
giving  minimum  current,  when  the  power  input  measured  by 
the  wattmeters  in  the  alternating-current  lines  plus  the  field 
losses  approximately  equals  P.  The  apparent  resistance  may 
be  obtained  by  measuring  the  actual  resistance  of  the  armature 
and  multiplying  by  the  ratio  of  apparent  to  true  resistance  for 
the  number  of  phases  of  the  machine  tested.*  From  this  the 
PR  losses  can  be  calculated.  Otherwise  tests  of  this  kind  of 
apparatus  conform  with  that  of  other  electrical  machines. 

The  efficiencies  of  converters  are  higher  than  the  efficiencies 

* Art.  178. 


SYNCHRONOUS  MACHINES 


781 


of  direct-current  or  alternating-current  generators  of  corre- 
sponding size,  on  account  of  the  greater  output  obtained  for  the 
converters  with  given  conditions  of  loss,  as  set  forth  in  Art.  178. 

When  converters  are  feeding  a three-wire  direct-current  sys- 
tem, the  neutral  conductor  of  the  direct-current  side  may  be 
connected  to  the  neutral  point  of  the  alternating-current  side. 
For  a quarter-phase  converter,  the  secondary  windings  of  the 
supply  transformers  can  be  connected  to  the  direct-current  neu- 
tral conductor  at  their  neutral  points.  For  a tri-phase  con- 
verter, the  neutral  point  can  be  readily  obtained  from  a wye  or 
tee  connection  of  transformers  and  for  a six-phase  converter  the 
connections  may  be  conveniently  either  wye  or  tee. 

182.  Inverted  Converters  : Double-Current  Generators.  When 
a converter  is  driven  by  direct-current  power  and  delivers 
alternating  current  to  the  line,  it  is  termed  an  inverted  rotary 
converter.  When  so  run,  the  machine  has  features  similar  to 
that  of  a direct-current  shunt  motor.  Thus,  the  speed  varies 
inversely  as  the  field  strength  and  directly  as  the  impressed 
voltage.  The  alternating-current  end  acts  like  an  alternating- 
current  generator,  a lagging  current  delivered  by  the  machine 
causing  a weakening  of  the  field  magnet,  and  a leading  cur- 
rent delivered  by  the  machine  causing  strengthening  of  the 
magnet.  Therefore,  if  an  inverted  rotary  finds  variable  alter- 
nating inductive  loads,  the  speed  and  hence  the  frequency  are 
apt  to  be  dangerously  variable.  It  is  therefore  common  to  use 
an  individual  shunt-wound  exciter  for  such  a machine,  which  is 
driven  at  a speed  proportional  to  that  of  the  inverted  converter. 
The  field  magnet  of  the  exciter  is  designed  with  low  magnetic 
saturation,  so  that  with  slight  increase  of  speed  the  exciter 
voltage  is  largely  increased,  thus  sending  an  increased  current 
through  the  inverted  converter  field  windings  and  thereby  main- 
taining approximately  normal  speech  If  the  armature  of  an  in- 
verted converter  has  little  reaction  on  the  field,  and  if  the  field 
magnet  is  highly  saturated,  very  little  trouble  need  be  experi- 
enced. Mechanical  and  other  devices  may  be  used  if  necessary 
to  control  the  speed.  The  distribution  of  current  among  the 
armature  conductors  and  other  characteristics  for  inverted  con- 
verters may  be  determined  in  the  same  manner  as  that  developed 
in  the  preceding  discussion  of  the  non-inverted  converter. 

In  the  use  of  inverted  converters,  and  indeed  in  all  alter- 


782 


ALTERNATING  CURRENTS 


nating-current  generating  plants,  it  is  desirable  to  have  a Fre- 
quency indicator  attached  to  the  lines.  Such  an  instrument  can 
be  made  on  the  principle  of  a rotary  field  induction  motor,  where 
a pointer  is  attached  to  the  armature,  which  is  restrained  by 
springs.  The  torque  of  the  armature  varies  as  the  frequency  and 
hence  a scale  in  frequency  can  be  made  for  the  pointer  to  travel 
over.  The  frequency  may  also  be  determined  by  an  instru- 
ment depending  upon  the  vibration  of  reeds  and  in  other  ways. 

Double-current  generators  are  sometimes  of  use  in  plants 
connected  to  both  direct  and  alternating  current  systems.  The 
double-current  generator  differs  from  the  converter  in  that  it 
is  driven  by  mechanical  power  and  therefore  both  the  alter- 
nating and  direct  currents  are  in  relatively  the  same  direction 
in  the  armature  windings.  If  curve  M in  Fig.  451  is  reversed, 
its  resultant  with  JSf  gives  the  current  distribution  in  the  arma- 
ture conductors  of  a double-current  generator  when  the  power 
factor  is  unity  and  equal  amounts  of  power  are  being  delivered 
to  the  direct  current  and  the  alternating-current  circuits.  It  is 
evident  from  the  construction  of  Fig.  451  that  reversing  curve 
M must  result  in  a relatively  great  I2R  loss,  and  a further  in- 
vestigation will  show  a larger  I2R  loss  for  a given  output  in  a 
double-current  generator  than  in  an  equivalent  converter  for 
the  same  output. 

183.  The  Alternating-Current  Motor-generator.  — The  motor- 
generator  consists  of  an  alternating  current  motor  driving  a di- 
rect current  generator  or  vice  versa , or  another  alternating  cur- 
rent generator,  the  two  machines  being  mounted  on  a common 
bed  plate  and  shaft.  In  the  latter  case  the  generator  in  commer- 
cial practice  is  usually  of  different  frequency  from  that  of  the 
motor,  and  the  machine  is  called  a Frequency  changer.  A problem 
of  importance  is  to  be  met  in  connecting  up  and  paralleling  the 
latter  machines  when  they  are  to  run  in  parallel  with  each  other. 
Suppose,  for  instance,  that  the  motor  of  a synchronous  motor- 
generator  set  is  fed  in  parallel  with  the  motors  of  other  motor- 
generators  with  currents  having  a frequency  of  25  cycles  per 
second,  and  the  generator  delivers  currents  in  parallel  with  the 
generators  of  the  same  motor-generator  sets,  the  currents  having 
a frequency  of  60  cycles  per  second.  The  number  of  poles  on 
the  two  machines  of  each  motor-generator  set  must  have  a ratio 
equal  to  the  ratio  of  their  respective  frequencies;  thus  if  the 


SYNCHRONOUS  MACHINES 


783 


25-cycle  motor  has  10  poles,  the  60-cycle  generator  must  have 
24  poles.  With  the  ratio  of  frequencies  incommensurable,  like 
25 : 60  = 5 : 12,  there  is  much  embarrassment  to  conjointly  meet 
the  mechanical  and  electrical  requirements  of  good  commercial 
machines.  Each  field  magnet  must  contain  an  even  number  of 
poles,  and  the  number  of  pairs  of  poles  must  be  in  the  ratio  of 
5 : 12.  This  gives  10  poles  and  24  poles  for  the  respective  field 
magnets  as  the  smallest  numbers  that  will  fit.  The  speed  of  the 
motor-generator  set  with  these  numbers  is  300  revolutions  per 
minute.  The  next  polar  relation  is  20  : 48,  which  gives  a speed 
of  150  revolutions  per  minute  and  no  exact  ratio  intervenes. 
The  choice  of  two  incommensurable  numbers  like  25  and  60  for 
the  standard  frequencies  therefore  causes  inconvenience  in  this 
situation.  A modification  to  24  and  60  would  improve  the  re- 
lations here  and  also  elsewhere. 

After  the  motor  is  running  properly  in  synchronism  and  the 
voltages  of  both  motor  and  generator  are  right,  the  generator 
phases  may  be  out  of  step  with  those  of  the  mains  to  which  it 
is  to  be  connected,  and  it  may  become  necessary  to  reverse  the 
polarity  of  the  motor  field  magnet  by  the  field  switch  and  pos- 
sibly continue  opening  or  reversing  the  field  circuit  a number 
of  times  in  order  that  the  motor  may  fall  back  successive  angles 
of  180  electrical  degrees  until  the  counter-voltages  in  the  motor 
and  the  terminal  voltages  in  the  generator  hold  the  proper  step 
relations  coincidently  in  their  respective  circuits ; or  the  re- 
quired step  relations  may  sometimes  be  obtained  by  reversing 
the  polarity  of  the  generator  field  magnet.  Even  when  the 
best  practicable  point  is  reached,  a large  synchronizing  current 
may  flow  if  the  motor-generator  does  not  have  quite  its  designed 
ratio  of  frequencies,  i.e.  25  to  60,  or  if  the  angular  relations  of 
the  parts  of  the  motor  and  the  generator  are  not  adjusted  to  ac- 
curately give  the  proper  time  phase  between  their  two  voltages. 

When  a motor-generator  set  is  being  brought  into  parallel 
operation  with  others,  it  must  be  brought  up  to  speed  (the 
motor  usually  starting  as  an  induction  machine)  and  the  motor 
be  connected  to  the  supply  circuit  when  the  indications  of  the 
synchroscope  are  favorable.  The  generator  is  then  in  syn- 
chronism, but  the  synchroscope  must  still  be  used  to  show 
whether  the  generator  is  in  proper  step  with  the  generator  bus 
bars  before  it  is  connected  to  its  circuit. 


CHAPTER  XII 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 

184.  Asynchronous  Motors  and  Generators  Defined.  — Motors 
called  Synchronous  are  those  in  which  the  moving  parts  rotate  at 
such  a velocity  with  reference  to  the  fixed  parts  that  an  angular 
distance  equal  to  that  of  the  polar  pitch  is  passed  over  during 
the  time  of  a half  cycle  of  the  impressed  alternating  voltage,  i.e. 
the  motor  runs  in  Synchronism  with  the  generating  apparatus 
which  drives  it,  and  it  has  the  same  speed  for  all  loads  pro- 
vided the  impressed  voltage  is  of  constant  frequency. 

Motors  which  do  not  have  this  characteristic,  but  vary  the 
relative  speed  between  the  fixed  and  movable  parts  more  or 
less  independently  of  the  frequency  of  the  impressed  voltage, 
are  called  Asynchronous  motors.  Alternating  current  Induction 
motors,  Repulsion  motors,  Series  motors,  and  others  in  which 
there  is  no  magnetic  field  set  up  by  direct  current  are  usually 
of  the  latter  class.  Asynchronous  generators  are  similar  to 
asynchronous  motors  in  possessing  no  definite  relation  between 
the  armature  speed  and  main  circuit  frequency. 

185.  Rotary  Field  Induction  Motors.  — The  well-known  prin- 
ciples which  cause  the  rotation  of  a disk  of  copper  pivoted 
above  a rotating  horseshoe  magnet  have  been  put  into  use 
through  the  discoveries  of  Ferraris,  Tesla,  Hasel wander,  Dob- 
rowolsky,  and  many  others.  The  arrangements  proposed  by 
Tesla  were  doubtless  the  first  direct  applications  of  these  prin- 
ciples to  commercial  use,  in  which  they  now  play  a primary  part 
in  the  transmission  and  distribution  of  power.*  An  almost 
simultaneous  publication  of  a series  of  scientific  experiments  by 
Ferraris  shows  the  operation  of  similar  apparatus,  f and  various 
experiments  of  a similar  nature  or  for  a similar  purpose  are  on 

* A New  System  of  Alternate-Current  Motors  and  Transformers,  Trans. 
Amer.  Inst.  E.  E.,  Vol.  5,  p.  308. 

t Electro-dynamic  Rotation  by  Means  of  Alternating  Currents,  London  Elec- 
trician, Vol.  21,  p.  86. 


784 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


785 


Fig.  457.  — Apparatus  for  setting 
up  a Rotating  Magnetic  Field. 


record.  Each  of  these  experiments  caused  an  iron  or  copper 
armature  to  rotate  when  placed  within  the  region  of  a rotating 
magnetic  field,  and  these  are  the  foundation  of  the  now  well- 
known  asynchronous  induction  motors. 

186.  A Rotating  Magnetic  Field.  — If  two  pairs  of  coils  are 
placed  at  right  angles  on  a laminated  iron  ring  (Fig.  457), 
with  the  connections  so  arranged  that  the  coils  of  each  pair  are 
in  magnetic  opposition  in  the  ring,  or  again,  if  the  two  pairs  of 
coils  are  placed  on  two  pairs  of  salient  poles,  or  embedded  in 
the  inside  face  of  the  ring,  so  that 
the  current  in  each  pair  of  coils 
tends  to  send  magnetic  flux  directly 
across  a central  core  (7,  the  magnetic 
flux  set  up  in  the  core  when  a cur- 
rent is  passed  through  the  coils  may 
be  considered  as  the  resultant  of  the 
magnetization  due  to  the  two  coils 
or  pairs  of  coils.  As  in  a transformer, 
if  the  resistance  of  the  coils  is  rela- 
tively small,  the  magnetic  flux 
threading  those  of  each  phase  when  alternating  currents  flow 
in  them  must  be  such  that  it  will  create  a counter- voltage 
in  the  coils  practically  equal  and  opposite  to  the  impressed 
voltage.  This  must  always  be  the  case,  whatever  may  be 
the  form  of  the  impressed  voltage  wave ; consequently,  if 
the  voltage  wave  is  sinusoidal,  the  wave  of  magnetic  flux 
within  the  coils  must  be  sinusoidal,  because  its  rate  of  change 
is  sinusoidal.  On  the  other  hand,  the  wave  shape  of  the 
exciting  current  takes  the  form  requisite  to  set  up  the  re- 
quired core  flux,  as  in  transformers,*  and  its  wave  form  is, 
therefore,  dependent  upon  the  variation  in  the  reluctance  of 
the  magnetic  core  caused  by  saturation  and  hysteresis,  f The 
resultant  flux  in  any  fixed  direction  across  the  central  core 
0 will  vary  in  value  as  in  the  common  core  of  a quarter-phase 
transformer,  but  the  special  arrangement  of  the  magnetic  cir- 
cuit causes  it  also  to  vary  in  direction  in  case  the  currents 
in  the  two  circuits  are  not  in  the  same  phase.  Considering 
a more  or  less  conventional  case  where  four  coils,  as  in  Fig. 
457,  are  supposed  to  uniformly  cover  the  whole  ring,  which  is 

t Art.  126. 


* Art.  118. 


78G 


ALTERNATING  CURRENTS 


equivalent  to  an  elementary  arrangement  in  a rotary  field  induc- 
tion motor  winding,  the  magnetic  reluctance  of  the  ring  and 
core  C being  assumed  constant  for  all  paths  the  flux  may  take 
parallel  to  the  plane  of  the  paper,  the  following  propositions 
are  approximately  true. 

If  two  equal  sinusoidal  voltages  90°  apart  in  phase  are  im- 
pressed upon  the  respective  equal  pairs  of  coils  on  the  ring  of 
Fig.  457,  the  exciting  currents  must  be  sinusoidal  to  create  the 
required  sinusoidal  magnetic  flux,  the  magnetic  reluctance  being 
assumed  to  be  constant,  and  the  magneto-motive  forces  must, 
therefore,  be  sinusoidal.  Under  these  conditions  at  any 
instant,  the  magnetizing  force  due  to  one  pair  of  the  coils  is 
H,  = Hm  sin  «,  and  the  magnetizing  force  due  to  the  other  pair 
of  coils  is, 

= Hm  sin  (a  — 90°)  = — H m cos  a, 


where  Hm  is  the  maximum  magnetizing  force  of  either  pair  of 
coils.  Thus  the  two  magnetizing  forces  differ  in  phase  by 
90°,  and  the  resultant  magnetizing  force  of  the  two  pairs  of 
coils  is  then  //,  = Vi/j2  + H 22  = Hm , as  illustrated  in  Fig.  458, 
and  is,  therefore,  constant  in  magnitude.  The  direction  in 
which  this  constant  magnetizing  force  acts  across  the  core  0 
varies  with  «.  When  a — 0°,  H,  tends  to  lie  in  the  plane  of  one 
pair  of  coils,  and  when  « = 90°,  it  tends  to  lie  in  the  plane  of 
the  other  pair  of  coils.  The  magneto-motive  force  of  each  pair 
of  coils  has  a sinusoidal  or  harmonic  variation,  and  the  resultant 
magnetizing  force  is  the  resultant  of  two  harmonic  variations 
with  90°  difference  of  phase.  Such  a resultant  tends  to  have  a 
uniform  magnitude  and  a uniformly  rotating  direction  in  one 
plane.  The  instantaneous  values  of  the  resultant  may,  there- 
fore, be  diagrammatically  represented,  with  approximate  ac- 
curacy under  the  conditions  assumed,  by  the  instantaneous 
positions  of  a line  of  fixed  length,  rotating  at  a uniform  rate 
around  one  end,  such  as  OHr  in  Fig.  458.  In  this  figure  the 
vector  of  magneto-motive  force  Hx  of  one  pair  of  the  coils  of 
Fig.  457  is  considered  to  lie  in  the  line  AB , and  to  vary  sinus- 
oidally in  magnitude,  while  the  magneto-motive  force  R2  of 
the  other  pair  of  coils  lies  in  the  line  CD , and  also  varies  sinus- 
oidally in  magnitude.  The  combination  of  these  two  magneto- 
motive forces  in  space  causes  the  resultant  Rr , which  as  ex- 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


787 


plained  is,  under  the  conditions,  approximately  constant  in  length 
and  of  uniform  coplaner  rotation.  In  Fig.  457  the  small 
arrows  show  the  direction  of  the  magnetic  fluxes  at  the  instant 
when  the  currents,  and  hence  the  magneto-motive  forces,  are 
equal  in  the  two  branches  of  the  two-phase  circuit.  As  the 
current  in  the  coils  of  one  phase  dies  out,  and  the  other  in- 
creases to  a maximum,  the  flux  in  C swings  around  45°  on  the 
center  of  C as  an  axis.  When  that  point  is  reached,  the  current 
is  zero  in  one  pair  of  coils,  and  the  other  pair  furnishes  the 
entire  flux  in  O.  As  the  current  now  rises  in  the  reverse 
direction  in  the  first  pair  of  coils  and  falls  in  the  other  pair, 
the  flux  in  C is  still  caused  to  rotate  by  virtue  of  the  rotating 
direction  of  the  resultant  magneto-motive  force.  In  this  way 
the  direction  of  the  flux  in  C uniformly  rotates  through  860° 
during  the  time  of  each  cycle  of  the  current.  This  may  be 
readily  illustrated  by  a series  of  diagrams  similar  to  those 
shown  in  Fig.  457,  in  which  the  magneto-motive  forces  are 
represented  by  arrows  for  the  range  of  angular  advances  during 
a cycle  of  the  currents.  The  arrows  must  be  reversed  in  direc- 
tion in  a pair  of  coils  when  the  current  reverses,  and  should  be 
made  approximately  proportional  in  length  to  the  instantaneous 
currents.  The  instantaneous  current  for  each  value  of  a may 
be  taken  from  a pair  of  sine  curves,  drawn  with  one  displaced 
90°  from  the  other.  Y 

If  the  maximum  value 
of  the  ampere-turns  of 
one  pair  of  coils  is  greater 
than  that  of  the  other 
pair  of  coils  instead  of 
the  two  being  equal,  as 
above  assumed,  the  mag- 
nitude of  the  resultant 
magnetizing  force  varies. 

The  rotating  field,  in  this 
case,  may  be  diagram- 
matically  represented  by 
a uniformly  rotating  line, 
which  varies  in  length 
so  that  its  tip  approxi- 
mately traces  an  ellipse 


Fig.  458.—  Locus  Diagram  of  a Uniform  Constantly 
Rotating  Magneto-motive  Force  created  by  Two 
Equal  Harmonically  Vibrating  Magneto-motive 
Forces  spaced  90  Mechanical  Degrees  Apart. 


788 


ALTERNATING  CURRENTS 


whose  minor  and  major  axes  are  respectively  in  the  planes  of 
the  stronger  and  weaker  coils.  If  the  windings  of  the  coils 
are  similar,  and  the  currents  equal,  but  the  phase  difference  is 
not  90°,  a variable  field  again  results. 

If  the  phases  of  the  two  currents  are  in  unison  instead  of  in 

quadrature,  l 

Hr  = VT^2  + H*  = V2  Hm  sin  «. 


This  shows  that  when  the  two  currents  are  in  unison  the  mag- 
nitude of  IIr  varies  with  sin «,  and  therefore  varies  from 
— V2  Hm  through  zero  to  + V2  Hm,  hut  the  direction  of  Hr  re- 
mains constant,  since  the  instantaneous  values  of  its  two  com- 
ponents are  always  equal.  Its  direction  evidently  lies  in  a line 
corresponding  to  the  arrows  in  Fig.  457  (or  90°  therefrom  de- 
pending upon  the  connections)  if  the  two  currents  are  equal, 
and  turned  more  or  less  from  that  position  if  the  two  currents 
are  unequal.  The  diagrammatic  representation  of  the  resultant 
here  is  a line  of  fixed  direction  which  harmonically  varies  in 
length,  the  total  range  of  variation  being  from  — V2  Hm  to 

+ V2 X- 

For  any  difference  of  the  current  phases  between  zero  and 

90°,  both  the  magnitude  and 
direction  of  Hr  vary,  and  the 
diagrammatic  representation 
is  a rotating  line  with  its  tip 
approximately  tracing  an  el- 
lipse. The  ratio  of  the  two 
axes  depends  upon  the  phase 
difference  of  the  currents. 
If  the  currents  have  90°  phase 
difference,  but  the  planes  of 
the  coils  are  not  90°  apart, 
the  effect  on  the  resultant 
magnetizing  force  is  evidentlv 
the  same  as  if  the  conditions 
were  reversed. 

If  the  impressed  voltages  are  not  sinusoidal,  the  value  of  the 
resultant  magnetizing  force  Hr  varies  in  a more  or  less  irregular 
manner,  inasmuch  as  the  different  harmonics  combine  in  differ- 
ent ways.  For  instance,  in  a quarter-phase  circuit  the  third 
harmonics  of  the  two  magneto-motive  forces  produce  a result- 


A' 


Fig.  459.  — Locus  of  Rotating  Field  Vector 
when  the  Exciting  Current  Wave  is  of 
Distorted  Form. 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


789 


/ 

1 

\ \ 

! o 

v 

//  / 

ant  that  rotates  in  the  opposite  direction  from  the  rotation  of 
the  resultant  produced  by  the  fundamentals,  and  the  rotation  is 
three  times  as  fast.  The  fifth  harmonics  produce  a resultant 
which  rotates  in  the  same  direction  as  the  resultant  produced  by 
the  fundamentals,  but  rotates  five  times  as  fast.  The  heavy  line 
in  Fig.  459  illustrates  the  locus  of  the  tip  of  the  rotating  field 
vector  produced  by  two  mag- 
neto-motive forces  in  time 
and  space  quadrature,  when 
each  consists  of  a funda- 
mental and  a third  harmonic, 
the  latter  located  to  cause 
peakedness  of  the  wave  of  B' 
magneto-motive  force.  Fig- 
ure 460  shows  a correspond- 
ing locus  when  the  third 
harmonic  causes  a flattening 
of  the  top  of  the  wave  of 
magneto-motive  force.  The 
value  of  flux  for  a = 0°,  90°, 

180°,  and  270°  is  alike  for  the 
two  figures,  and  the  dotted  circles  show  the  locus  for  sinusoidal 
magneto-motive  forces. 

The  same  argument  may  be  readily  seen  to  apply  to  the  re- 
sultant magnetizing  force  due  to  any  number  of  coils  surround- 
ing a core.  Thus,  an  arrangement  similar  to  that  of  Fig.  457, 
except  that  it  is  applicable  for  obtaining  a rotating  field  from 
a tri-phase  circuit,  may  be  constructed  by  winding  three  coils 
upon  the  ring  and  property  connecting  them  to  the  three  phases 
of  a triphase  supply  circuit.  When  equal  coils  are  at  equal 
angular  distances,  and  equal  currents  in  the  individual  coils 
differ  in  phase  by  an  amount  equal  to  the  angular  distances  of 
the  coils  from  each  other,  the  resultant  magnetizing  force  is 
always  approximately  uniform  in  magnitude  under  the  condi- 
tions assumed,  and  rotates  at  a uniform  rate,  provided  the 
currents  are  sinusoidal.  The  magnitude  of  the  resultant  is 

ib  In-iF;, 

Hr—  2 Hm,  where  m is  the  number  of  phases,  in  accordance 

with  the  theorems  of  simple  harmonic  motion.  The  correctness 
of  these  deductions  has  been  proved  by  experiment. 


Fig.  460.  — Locus  of  Rotating  Field  Vector 
when  the  Exciting  Current  Wave  is  of  Dis- 
torted Form. 


790 


ALTERNATING  CURRENTS 


The  Germans  call  the  rotating  magnetic  field  Drehfelde,  and 
the  polyphase  currents  which  set  up  a rotating  magnetic  field 
the  Drehstrom,  or  rotating  current.  In  the  actual  rotating  field, 
the  magnetic  flux  does  not  necessarily  vary  with  absolute  con- 
stancy in  accord  with  the  propositions  given  above,  but  it  varies 
even  if  the  windings  are  uniformly  distributed.  This  variation 
is  dependent  upon  the  variation  of  the  instantaneous  values  of 
magneto-motive  force  and  reluctance  of  each  elementary  width 
of  the  magnetic  circuit  through  the  central  core.  Indeed,  it  is 
possible  for  the  resultant  magnetic  flux  to  vary  in  instantaneous 
value  as  much  as  40  per  cent,  as  it  rotates,  even  in  the  case  of 
properly  distributed  coils  and  balanced  voltages,  when  there  is 
no  closed  secondary  circuit  acted  on  by  the  flux.  The  extent 
of  this  variation  depends  upon  the  number  of  phases  of  the 
windings  and  other  variables.  However,  when  secondary  cir- 
cuits on  the  central  core  are  closed,  the  reactions  are  found  to 
smooth  out  the  irregularities  that  occur,  and  the  wave  of  result- 
ant magnetic  flux  may  be  considered  approximately  sinusoidal 
in  distribution  and  of  approximately  constant  value  in  magni- 
tude and  rate  of  rotation  in  those  cases  in  which  care  has  been 
exercised  to  properly  place  the  windings  on  the  core,  and  when 
approximately  sinusoidal  impressed  voltages  are  used. 

187.  Action  of  a Short-circuited  Armature  Winding  within  a 
Rotating  Field.  — If  a drum  core  of  laminated  iron  is  suitably 
pivoted  within  a ring  on  which  coils  carrying  alternating  cur- 
rents are  so  situated  that  the  resultant  magnetic  field  rotates, 
the  pivoted  core  will  be  dragged  into  rotation  by  the  magnetic 
pull.  If  the  pivoted  core  is  a cylinder  of  copper,  it  will  be 
dragged  into  rotation  by  the  reactions  between  the  rotating 
magnetic  flux  and  the  eddy  currents  which  are  developed  in 
the  cylinder.  The  latter  is  directly  analogous  to  the  classic 
experiment  of  Arago  with  a disk  of  copper  pivoted  before  the 
poles  of  a rotating  permanent  magnet. 

In  the  case  of  either  a solid  core  or  Arago  disk,  the  eddy 
currents  are  not  constrained  in  position  and  therefore  take  the 
paths  of  least  resistance.  The  result  is  that  much  of  the  effec- 
tiveness of  the  currents  in  bringing  about  a rotation  is  lost,  and 
the  efficiency  of  the  device  is  small.  If,  in  the  disk  experi- 
ment, the  disk  is  cut  up  into  an  indefinitely  large  number  of 
fine  radiating  wires  which  are  connected  together  at  their  inner 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


791 


and  outer  ends,  the  losses  due  to  parasitic  eddies  may  in  a large 
measure  be  done  away  with,  and  the  efficiency  of  the  device  is 
considerably  raised.  In  the  same  way  the  drum  core  may  be 
made  of  laminated  iron  in  order  that  the  magnetic  circuit  shall 
be  of  small  reluctance,  and  embedded  in  this  may  be  copper 
wires  which  cross  the  face  of  the  core  and  are  all  short- 
circuited  by  copper 
rings  at  the  ends 
(Fig.  461),  making  a 
cage  of  conductors 
like  a squirrel  cage. 

These  make  con- 
strained paths  for 
the  induced  currents, 
and,  if  the  core  is 
Sufficiently  laminated  FI(j.  Til.  — Induction  Motor  Armature  having  a Short- 
, ,i  circuited  (Squirrel  Cage)  Winding. 

and  the  copper  con- 
ductors are  not  too  thick,  the  parasitic  eddies  are  largely  done 
away  with  and  the  efficiency  of  such  a motor  may  be  made  quite 
large.  This  construction  is  the  essential  construction  of  what 
are  known  as  induction  motors. 

There  has  been  more  or  less  ambiguity  in  applying  the  desig- 
nations of  field  and  armature  windings  to  the  primary  and  sec- 
ondary windings  of  induction  motors,  since  either  may  rotate, 
and  both  windings  carry  alternating  currents.  The  following 
definitions  avoid  all  ambiguities  in  simple  machines.  The  Field 
magnet  is  the  core  upon  which  are  placed  windings  connected  to 
the  external  circuit.  The  currents  in  the  field  windings  are, 
therefore,  due  to  the  impressed  voltage  of  the  external  circuit. 
The  Armature  is  the  part  carrying  conductors  in  which  current 
is  induced  by  the  revolving  magnetism  of  the  field  magnet. 
Since  the  current  is  inductively  set  up  in  the  armature  conductors, 
these  motors  are  called  Induction  motors.  It  is  readily  seen  that 
the  induction  motor  is  a transformer  as  well  as  a motor,  the  pri- 
mary winding  of  which  is  on  the  field  magnet,  and  the  secondary 
winding  on  the  armature,  and  it  is  convenient  to  call  the  windings 
receiving  power  conductively  from  the  power  supply  circuit  the 
Primary  windings,  and  those  in  which  current  is  induced  the 
Secondary  windings.  The  part  that  rotates  in  any  alternating 
current  machine  having  a rotating  part  is  often  called  the 


792 


ALTERNATING  CURRENTS 


Rotor,  while  that  which  is  stationary  is  called  the  Stator,  irre- 
spective of  which  is  the  primary  or  secondary  part. 

188.  Counter-voltage  induced  in  Primary  Circuit.  Exciting 
Current.  — Treating  the  primary  winding  of  an  induction  motor 
in  the  same  manner  as  the  primary  winding  of  a transformer, — 
which  is  permissible,  — the  following  formula*  gives  the  rela- 
tion between  counter-voltage,  frequency,  magnetism,  and  the 
turns  of  the  coils  : 

E' 

1 1 A8 


in  which  E\  is  the  induced  voltage  in  the  coil,  nx  the  number 
of  turns  in  the  coil,  $ the  maximum  flux  including  the  leakage 
flux  from  one  pole  of  the  magnetic  field,  f the  frequency  of  the 
magnetic  cycles,  provided  all  the  magnetism  is  included  within 
all  the  turns  of  the  winding.  This  proviso,  however,  must  be 
modified  in  an  ordinary  induction  motor,  since  the  magnetic 
density  in  the  air  gap  may  be  assumed  to  vary  as  a sinusoid,! 
and  that  condition  requires  that  the  number  of  lines  of  force 
passing  through  the  different  turns  of  the  coils  shall  also  vary 
as  a sine  function.  This  is  illustrated  in  Fig.  462.  In  this 
figure,  overlapping  distributed  polyphase  windings  are  laid  in 
slots  in  the  internal  face  of  the  field  magnet  ; and  polyphase 
currents  of  a corresponding  number  of  phases  flowing  in  the  coils 
produce  a resultant  rotating  magnetic  flux  distribution  in  the 
air  gap  between  field  magnet  and  armature  similar  to  that 
shown  in  the  figure  for  a particular  instant  by  the  dotted  lines 
in  the  air  gap.  The  figure  pertains  to  a four-pole  field  magnet. 

Secondary  windings  are  laid  in  slots  in  the  external  surface 
of  the  cylindrical  armature  core,  and  both  primary  and  second- 
ary windings  are  laid  down  so  as  to  uniformly  cover  the 
winding  surface  of  the  core  on  which  it  is  laid,  as  is  usual  in 
commercial  practice.  The  broken  lines  in  the  polar  space  and 
the  curves  indicate  the  approximate  magnetic  distribution. 
Consequently  we  have  for  the  induction  motor, 


- 1 I cos  uda 

1 10s  6 


e . 


where  ^ is  the  value  of  « corresponding  to  the  sine  ordinate 


* Art.  135. 


t Art.  186. 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


793 


which  is  proportional  to  the  number  of  lines  of  force  passing 
through  the  extreme  turns  of  the  winding  concerned,  when  the 
center  of  the  magnetic  field  is  exactly  over  the  center  turns  of 
the  coil. 


Fig.  462. — Diagram  showing  Primary  and  Secondary  Windings  of  an  Induction 
Motor  evenly  distributed  over  Internal  aud  External  Surfaces  of  Field  Magnet 
and  Armature  Core  respectively,  and  also  showing  the  Approximate  Distribu- 
tion of  Magnetic  Flux  for  a Given  Instant. 


For  uniformly  distributed  windings  in  two  phases,  9 = 90°; 
and  in  three  phases,  9 = 60° ; while  for  coils  on  salient  poles 
9—  0°.  The  values  of  E'  for  the  field  winding  of  the  induction 
motor  then  become 


4 n^f 


E'  „ = and  E'  , 

1,2  L3 


_ 3V2  n&f 


108 


where  E\  2 is  the  induced  voltage  in  the  coils  of  one  phase  of 
a quarter-phase  motor  field  winding  and  E\  3 is  the  induced 
voltage,  likewise,  for  a tri-phase  motor  field  winding.  Hence, 
the  voltage  set  up  in  a uniformly  distributed  field  winding  of  a 


794 


ALTERNATING  CURRENTS 


quarter-phase  motor  is,  other  things  being  equal,  about  10  per 
cent  less  than  if  the  windings  were  in  narrow  coils  ; and  in  a 
tri-phase  motor  the  deficit  is  nearly  5 per  cent.  The  exact 
ratios  are  7 r : 2v/2  and  7r  : 3.  To  give  the  same  counter  vol- 
tage in  the  uniformly  distributed  field  windings  of  an  induction 
motor  arranged  for  two  phases,  requires  about  6 per  cent  more 
turns  in  the  windings  than  when  the  same  machine  is  arranged 
for  three  phases.  If  the  windings  are  placed  on  salient  poles, 
as  was  at  one  time  done  in  quarter-phase  motors,  all  the  lines 
of  force  pass  through  the  windings,  and  the  coils  therefore  act 
as  though  they  were  very  narrow;  but  the  increased  and  irregu- 
lar reluctance  of  the  magnetic  circuit  caused  bjr  this  construc- 
tion more  than  destroys  any  advantage  for  ordinary  motors  per- 
taining to  this  form  of  the  winding. 

The  formulas  given  above  may  also  be  derived  by  consider- 
ing the  primary  windings  as  stationary  windings  of  an  alternat- 
ing-current generator  with  rotating  field  magnet  in  which  the 
magnetic  flux  from  the  poles  has  approximately  a sinusoidal 
distribution.  For  some  purposes  this  method  is  useful.*  It 
is  evident  that  the  voltage  set  up  in  the  windings  by  the  rotat- 
ing magnetism  is  exactly  the  same  whether  the  magnetic  flux 
moves  with  the  field  core  as  in  a synchronous  machine  or  moves 
through  the  core  by  reason  of  the  varying  resultant  of  two  or 
more  magneto-motive  forces  as  in  the  induction  motor.  The 
voltage  induced  in  a conductor  depends,  in  other  words,  upon 
the  relative  angular  velocity  of  conductor  and  magnetic  flux, 
and  it  makes  no  difference  how  the  relative  velocity  is  ob- 
tained. Therefore,  the  generator  formulas  apply  to  this  case,  of 


2 7r/q<I>  VP 
108  x 60 


sin  a,  f 


in  which  e\  is  the  instantaneous  induced  voltage  in  the  coil,  V 
the  revolutions  per  minute,  « the  instantaneous  angular  position 
of  the  coil  in  the  magnetic  field,  and  the  flux  per  magnet 
pole.  In  the  case  under  consideration, 

^ _ 2 2 7 rrlcf)  _ 2 rl<f> 

o ’ 

7T  2|)  p 

* Art.  20. 

t Blondel,  Notes  sur  la  tlieorie  dl^mentaire  des  appareils  k champ  tournant 
La  Lumiere  Vlectrique,  Vol.  5,  p.  351 ; Jackson,  Three-phase  Rotary  Field. 
Electrical  Journal,  Vol.  1,  p.  185. 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


795 


where  r and  l are  the  inner  radius  of  the  field  core  and  its 
length  parallel  to  the  motor  shaft,  <£  the  maximum  magnetic 
density  in  the  air  space,  and  p the  number  of  pairs  of  poles  in 

V f 

the  magnetic  field.  Since  — = — , and  the  magnetism  per 

60  p 

magnetic  circuit  in  a multi-polar  machine  must  be  multiplied 
by  the  number  of  pairs  of  poles  to  get  the  total  number  of  lines 
of  force  cut  per  conductor  per  revolution,  the  formula  may  be 
written,  generally,  in  the  form 


e\  = 


_ 2 Trnp$>f 


108 


sin  a ; 


whence 


e'm  — 


_ 2 7T»  p\)f 


108 


•> 


and 


E\  = 

1 1IJS 


which  is  as  in  the  transformer.  This  is  the  value  of  the  vol- 
tage developed  in  a narrow  coil,  but  if  the  coil  is  spread  over 
a considerable  area,  the  maximum  voltage  is  less,  as  shown 
earlier  in  the  article. 

For  various  reasons  it  is  more  convenient  to  stud}'  induction 
motors  from  the  transformer  standpoint,  and  we  may  consider 
them  as  transformers  with  relative  motion  between  the  pri- 
mary and  secondary  windings. 

Since  an  air  space  must  be  made  in  the  magnetic  circuit  to 
allow  such  a motor  to  operate,  it  is  evident  that  the  quadrature 
element  of  the  exciting  current  of  induction  motors  must  be  ma- 
terially greater  than  that  of  ordinary  transformers.  In  fact, 
the  no  load  current  of  some  comparatively  small  motors  of  this 
type,  which  show  quite  a high  full  load  efficiency,  is  entirely 
comparable  to  the  full  load  current.  To  reduce  the  quadrature 
current  to  a reasonable  limit,  every  effort  must  be  bent  to 
decrease  the  reluctance  of  the  air  space.  As  the  armature 
conductors  are  embedded  in  the  primary  and  secondary  cores, 
it  is  possible  to  make  the  air  space  simply  that  required  for 
mechanical  clearance ; and,  by  care  in  the  workmanship,  this 
may  be  made  quite  small  compared  with  the  air  space  of 
dynamos  built  according  to  the  ordinary  methods. 


796 


ALTERNATING  CURRENTS 


The  exciting  current  for  an  induction  motor  may  be  calcu- 
lated for  each  circuit  in  exactly  the  same  manner  as  that  for  a 
transformer.  It  is  composed  of  two  components  in  quadrature : 

(1)  The  active  component,  which  is  equal  to  the  sum  of  the  no- 
load  losses  in  the  circuit  measured  in  watts,  divided  by  the  volts 
per  circuit.  The  fixed  losses,  as  in  the  transformer,  vary  but 
little  from  no  load  to  full  load  in  well-designed  motors.  The 
number  of  circuits  is  equal  to  the  number  of  phases.  The 
total  losses  entering  into  the  exciting  current  (or  the  no  load 
losses)  are  the  core  losses  in  the  field  magnet  and  armature ; 
the  I2R  loss  in  the  field  winding  which  is  due  to  the  exciting 
current ; a small  I2R  loss  in  the  armature  conductors  which  is 
due  to  the  secondary  current  required  to  run  the  armature 
against  friction  and  armature  core  losses  ; and  the  friction  loss. 
The  watts  represented  in  the  exciting  current  per  electrical 
circuit  are  equal  to  the  total  no  load  losses  divided  by  the  number 
of  phases. 

(2)  The  magnetizing  ampere-turns,  which  are  in  quadrature 
with  the  active  component  of  the  exciting  current,  may  be 
calculated,  as  in  the  case  of  transformers,  from  the  formula 


nl— 


1.25 


, where  P is  the  reluctance  of  the  magnetic  circuit 


and  nl  represents  the  resultant  ampere-turns.  But  the  mag- 

2 

neto-motive  force  per  phase  is  equal  to  — times  the  resultant 

m 

magneto-motive  force.*  Therefore,  if  nx  is  the  number  of 
turns  per  phase  which  link  each  magnetic  circuit,  the  actual 
magnetizing  component  of  current  per  circuit  is 


T Px$  2 

Ifj.  — “=  X , 

rij  x V2  x 1.25  m 

where  m is  the  number  of  phases.  Which  for  a quarter-phase 
machine  becomes  approximately 


T _ P4> 

"2~  1.75  V 

and  for  a tri-phase  machine,  approximately 

T = P($> 

^ 2.65  nx 


* Art.  186. 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


797 


Since  mechanical  clearance  between  the  armature  and  field  mag- 
net is  an  essential  feature  of  a motor,  the  reluctance  of  the  motor 
magnetic  circuit  is  much  greater  than  that  of  a transformer  of 
corresponding  capacity.  This  makes  the  magnetizing  current 
greater,  as  stated,  increases  the  exciting  current,  and  reduces 
the  power  factor. 

The  total  exciting  current  is  equal  to  the  square  root  of  the 
sum  of  the  squares  of  its  two  components,  or  Ie  — V Ic2  + i^2, 
where  Ic  is  the  active  component  of  the  current. 

The  ampere-turns  on  each  magnetic  circuit  of  an  induction 
motor  are  the  resultant  of  the  ampere-turns  due  to  all  the 
phases.  It  is  therefore  evident,  from  previous  deductions,  that 
the  resultant  ampere-turns  in  the  magnetic  circuit  of  an  induc- 
tion motor  is  --(nl^ 2),  where  n is  as  before  the  number  of 

turns  belonging  to  each  phase  which  link  each  magnetic  circuit, 
m is  the  number  of  phases,  and  Ie  is  the  no  load  or  exciting 
current  in  each  phase.*  Consequently  if  y ampere-turns  are 
required  in  the  magnetic  circuit,  the  winding  in  each  phase 
2 

must  furnish  — times  the  whole,  as  was  assumed  earlier  in  the 
to 

paragraph  in  obtaining  a value  for  1^.  For  quarter-phase 

2 2 2 

motors,  — = 1,  and  for  tri-phase  motors,  — = -• 
to  mo 

189.  Motor  Speeds,  Slip,  and  Secondary  Induced  Voltage.  — 

The  velocity  of  rotation  of  the  resultant  primary  magnetic  flux 
depends  upon  the  frequency  of  the  current  supplied  to  the 
motor,  and  the  number  of  pairs  of  poles  in  the  field.  In  two- 
pole  machines,  the  number  of  rotations  which  the  magnetic 
flux  makes  per  second,  or  the  Field  frequency,  is  equal  to  the 
frequency  of  the  primary  voltage  ; and,  in  multi-polar  machines, 
the  field  frequency  is  equal  to  the  frequency  of  the  primary 

f 

voltage  divided  by  the  number  of  pairs  of  poles,  or  — . The 

P 

number  of  pairs  of  poles  which  is  referred  to  by  the  symbol  p is 
the  number  in  the  rotating  magnetic  field.  This  is  equal  to  the 
number  of  pairs  of  poles  set  up  by  the  windings  in  the  primary 

cores  with  a smooth  magnetic  surface,  but  is  equal  to  times 


* Art.  186. 


798 


ALTERNATING  CURRENTS 


the  number  of  polar  projections  which  would  be  necessary  if 
each  phase  coil  was  wound  upon  a projection,  though  in  this 
latter  case  the  number  of  resultant  magnet  poles  is  the  same  as 
where  the  smooth  core  is  used. 

The  velocity  of  rotation  of  the  secondary  winding  of  an 
induction  motor  can  never  equal  the  velocity  of  rotation  of  the 
rotating  magnetic  flux  (when  the  machine  is  running  as  an 
induction  motor),  since  the  secondary  conductors  must  be  cut 
by  the  lines  of  force  of  the  rotating  flux  in  order  that  voltage 
may  be  developed  in  the  armature  conductors  ; that  is,  the 
rotating  magnetic  field  must  always  have  a relative  velocity  of 
rotation  with  reference  to  the  armature  conductors.  In  any 
machine,  the  relative  velocity  given  in  terms  of  revolutions  per 
minute  is  v—V—  I77,  where  V and  V are  respectively  the 
number  of  revolutions  per  minute  of  the  magnetic  field  and  the 
secondary  conductors.  This  relative  velocity  measured  as  a 
fraction  of  the  rotating  field  velocity  is  called  the  secondary 
Slip,  and  is  small,  seldom  exceeding  5 or  10  per  cent  of  the 
speed  of  the  motor  for  ordinary  so-called  constant  speed 
machines,  and  reaching  these  values  only  in  small  motors. 
Slip  is  usually  for  convenience  named  as  a fraction  or  per- 
centage of  the  synchronous  velocity  V oi  the  machine,  in  which 

case  slip  is  s2  = -p ; and  as  in  a machine  with  a given  number  of 

poles,  the  frequency  of  the  current  in  the  primary  conductors 

must  be  to  the  frequency  in  the  secondary  conductors  as  — * 

f V 

therefore  s2  = where  f and  /2  are  the  two  frequencies,  re- 
spectively. Since  the  current  in  the  secondary  winding  must 
be  proportional  to  the  work  done  by  the  motor,  it  must  vary 
with  the  load,  if  the  secondary  impedance  is  constant,  and  v 
must  increase  as  the  load  is  increased.  A little  consideration 
shows  that,  if  the  rotating  field  flux  remains  constant  in  value, 
the  variation  of  v with  the  load  must  be  just  sufficient  to  coun- 
terbalance the  drop  of  the  secondary  voltage  caused  by  the 
current  flowing  in  the  secondary  conductors. 

A variation  of  v demands  a variation  of  V of  equal  magni- 
tude, since  J^is  fixed  by  the  frequency  of  the  current  delivered 
to  the  motor  ; consequently,  the  speed  regulation  of  a rotary 
field  motor  is  directly  dependent  upon  the  loss  of  voltage  in  the 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


799 


secondary  conductors,  if  we  neglect  the  effect  of  variable  arma- 
ture reactions  and  drop  of  voltage  in  the  primary  windings. 
This  is  analogous  to  the  case  of  direct-current  shunt -wound 
motors. 

At  starting,  the  relative  velocity  of  the  rotating  field  flux 
with  respect  to  the  secondary  conductors  is  evidently  V,  since 
V is  zero.  The  secondary  current  therefore  may  be  very 
great,  and  the  starting  torque  may  also  be  great  provided  the 
armature  reactions  do  not  too  greatly  disturb  the  phase  posi- 
tion of  the  rotating  magnetic  flux.  To  avoid  injury  to  the 
secondary  windings,  in  large  motors,  by  the  current  at  starting, 
means  must  be  taken  to  prevent  its  becoming  excessive,  exactly 
as  in  the  case  of  direct-current  motors  worked  on  constant 
voltage. 

From  the  relation  V — F7  = v,  it  is  evident  that  the  value  of  v 
determines  the  frequency  with  which  the  rotating  magnetic  flux, 
<F,  cuts  the  secondary  conductors,  and  it  will  vary  from  a value 
v = V to  v = 0 as  the  motor  armature  changes  from  a condition 
of  rest  to  a speed  of  synchronism.  Slip  is  frequently  named 
in  per  cent  of  motor  speed.  In  ordinary  practice,  the  slip 
varies,  at  full  load,  from  a fraction  of  1 per  cent  to  10  per 
cent  of  V,  having  the  smaller  value  in  large  machines. 

As  in  a transformer,  the  voltage  induced  in  the  turns  of  the 
secondary  winding  when  at  zero  speed  (7r=  0)  by  the  mutual, 
rotating  flux,  is  the  same  per  turn  as  that  induced  per  turn  in 
the  primary  winding,  and  its  value  is 

r,  V2  t r An^f 

^2-  1(J8 

where  <1?  is  the  flux  passing  through  botli  primary  and  secondary 
coils,  i.e.  the  mutual  flux  per  magnetic  circuit,  and  A.  is  a 
factor  depending  on  the  width  of  the  coils  per  phase,  being 
unity  for  very  narrow  coils  and  less  than  unity  when  the  coils 
are  distributed  over  material  width,  as  explained  in  the  previous 
article.  w2  is  the  number  of  secondary  turns  and  f the  frequency 
of  the  primary  supply  circuits.  In  a squirrel  cage  armature 
the  conductors  within  twice  the  width  of  the  polar  pitch  may 
be  conveniently  considered  as  composing  one  turn  which  is 
wound  of  parallel  conductors  distributed  over  180  electrical 
degrees  of  the  core  surface  and  wound  under  two  contiguous 


800 


ALTERNATING  CURRENTS 


poles  of  opposite  sign.  For  this  case  A becomes  equal  to  — . 

7T 

Under  these  circumstances  the  formula  becomes 

-pi  2V2  n2<bf 

E>=  10»  • 

For  other  armature  speeds  than  zero,  the  voltage  evidently  be- 
comes s2Ev  as  the  rate  at  which  the  secondary  conductors  are 
cut  by  the  rotating  field  is  proportional  to  s2,  i.e.  at  standstill 
the  secondary  voltage  is  proportional  to  the  speed  V,  while 

when  running  it  is  proportional  to  the  speed  v,  but  s0=  — , 

hence  for  any  speed  the  secondary  voltage  is  s2E2.  E2,  it  will 

be  noted,  is  the  standstill  secondary  voltage. 

It  is  sometimes  more  convenient  to  know  the  voltage  for  a 
conductor,  rather  than  the  resultant  voltage  for  all  the  conduc- 
tors in  series  or  parallel;  in  such  case  the  instantaneous  maxi- 
mum voltage  is,  for  any  slip  s2, 

_ 2 7rd>/s2  _ 2 TrA>pv 

e‘lmc-  108  ~ 1Q8  x 60’ 

where  <E>  is  the  maximum  value  of  the  mutual  rotating  flux  per 
pole.  The  effective  voltage  per  conductor  is 

p _ V2  7rfl>/s9  _ V2  irA>pv 
2c  ~ 108  ~ 108  x GO  ’ 

since  the  voltage  curves  in  the  conductors  must  he  sinusoidal 
if  the  magnetism  has  a sinusoidal  distribution  in  the  air  space 
and  the  velocity  is  uniform,  as  has  been  assumed. 

190.  Currents,  Torque,  Impedance,  and  Magnetic  Leakage  in  a 
Rotary  Field  Induction  Motor.  — The  reactance  of  an  induction 
motor  is  that  due  to  the  magnetic  leakage  between  the  primary 
and  secondary  windings,  as  in  a transformer.  The  reactance 
due  to  one  phase  of  the  secondary  windings  when  the  armature 
is  at  a standstill  and  the  slip  equals  unity  is 

X2  = 2 7 rfLv 

where  L2  is  the  self-inductance  of  the  secondary  winding  due  to 
leakage  flux  ; and  where  s2  equals  any  fractional  value  other 
than  unity,  X'2  = 2 i rf2Lv  but  f2  = s2/,  and 

X\  - s.2X2  = 2 t rfL2s2. 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


801 


Therefore,  the  impedance  of  the  secondary  winding  for  any 
slip  is 

^2  = ( + S22X22)K 

where  B2  is  the  resistance  of  the  secondary  winding.  The  cur- 
rent which  flows  in  the  secondary  winding  is  then 

J __  S2^2  _ ,S2^2 TT-An2<j>fs2 

2 ^2  mZ2 

The  symbols  for  current,  resistance,  reactance,  impedance, 
etc.,  have  the  same  subscripts  and  meaning  as  used  when  in 
dealing  with  the  secondary  windings  of  a transformer,  but  it 
must  be  remembered  that  the  impedance  Z2  varies  as  the 
speed  of  the  motor  changes. 

The  primary  current  has  a component,  of  which  the  magneto- 
motive force  is  equal  and  opposite  to  that  of  the  secondary 
current,  as  in  a transformer  ; therefore,  the  primary  current  is 
composed  of  the  exciting  current  and  the  component  required 
to  neutralize  the  mutual  magneto-motive  force  of  the  secondary 
current. 

The  angle  of  lag  in  the  secondary  circuit  is 


= tan-1-2— 2 = cos 
2 Bo 


Bo 


VB*  + s*  X* 


The  torque  between  the  field  magnet  and  armature  is  pro- 
portional to  the  integrated  product  of  the  rotating  mutual  mag- 
netic flux  with  the  secondary  current,  or 


T = KQIo  cos  6o  = jSTI? 


Bo 


V BJ  + s’zx*  V M2  -is  2X2, 


where  K depends  upon  the  construction  of  the  machine  and 
the  distribution  of  the  flux  and  current  at  the  surface  of  the 
armature.  From  this  the  torque  is  found  to  be 


T=K<t> 


So  Bo  B .) 

B*Vs*X*' 


From  the  formulas  given  it  is  seen  that  the  I2B2  loss  is 


„ 27^2 

P — b2  ^2 7?  - 

' B*  + S*X, 2 2 


Ts2E2 

K<$> 


3 F 


802 


ALTERNATING  CURRENTS 


Hence  there  results 


when  K can  be  considered  to  be  constant,  since  E2  is  a direct 
function  of  <t>. 

From  the  last  formula  it  is  seen  that,  if  the  motor  is  attached 
mechanically  to  a load  having  a constant  resisting  moment,  the 
copper  loss  and  slip  will  bear  the  same  ratio  whatever  may  be 
the  changes  in  the  secondary  resistance  or  the  primary  voltage. 
Thus,  if  the  secondary  resistance  is  lowered,  the  armature  speed 
increases  slightly  and  the  slip  becomes  less,  the  current  becomes 
somewhat  less,  but  the  cosine  of  the  angle  of  lag  decreases  pro- 
portionately so  that  the  electro-magnetic  torque  continues  to 
just  balance  the  mechanical  torque  of  the  load. 

In  a short-circuited  secondary  winding,  s.2E2  volts  are  all  used 
in  driving  the  current  through  its  impedance.  The  electrical 
power  converted  into  mechanical  power  by  the  motor  can  be 
expressed  by  S2E2I2  cos  d2  = E2I2  cos  d2  — cos  02,  when 

>%e2  is  a voltage  that  would  be  generated  if  the  armature 
ran  at  a slip  of  V — v = V'  and  S2  = 1 — s2.  An  analogous 
situation  is  created  if  the  rotating  flux  is  imagined  to  be  station- 
ary as  in  an  alternator  with  rotating  armature  and  the  arma- 
ture is  mechanically  driven  at  a speed  V ; then  there  results  a 
voltage  S2E2  generated  in  the  secondary  windings  if  E2  is  the 


voltage  accompanying  a speed  V and  *S2  = — , as  in  the  foregoing. 


In  the  case  of  the  motor  the  power  delivered  to  the  shaft 
equals  the  torque  times  the  velocity  of  rotation  in  angular  travel 
per  unit  of  time,  so  that  the  mechanical  power  in  watts  is  P 

= 2 7T  T.  Transposing  gives  T=  2 7 The  voltage  S2E2 

does  not  actually  appear  in  the  motor  as  an  electrical  quantity, 
but  as  a mechanical  torque,  and  it  is  represented  in  the  primary 
circuit  by  the  voltage  drop  whose  active  component,  multiplied 
by  the  component  of  primary  current  flowing  because  of  the  sec- 
ondary current,  equals  the  motor’s  mechanical  output  in  watts. 

In  a transformer  the  secondary  external  circuit  provides  a 
path  for  a current  which,  by  setting  up  a counter-magneto- 
motive force,  causes  a current  of  equal  and  opposite-magneto- 
motive force  to  flow  in  the  primary  windings.  The  vector 


ASYNCHRONOUS  motors  and  generators 


803 


product  of  this  current  by  the  impressed  voltage  equals  the 
transformer  output  plus  the  copper  losses  of  this  component  of 
current.  In  an  induction  motor  the  resisting  moment  of  the 
load,  by  lowering  the  velocity  of  rotation  of  the  armature,  is 
the  cause  of  the  flow  of  a current  in  the  secondary  windings 
which,  as  in  a transformer,  demands  current  of  equal  and  op- 
posite magneto-motive  force  in  the  primary  windings. 

191.  Vector  Relations  in  the  Rotating  Field  Induction  Motor.  — 
From  the  preceding  discussions  it  is  evident  that  a diagram  of 
current  and  voltage  loci  may  be  drawn  for  an  induction  motor 
similar  to  the  corresponding  diagram  for  a transformer.  In 
the  transformer  diagram  showing  the  effect  of  internal  reactance 
and  a non-reactive  load  (Fig.  277),  the  secondary  current  locus 
is  drawn  upon  the  basis  of  a circuit  containing  fixed  inductive 
reactance  and  variable  resistance.*  In  the  rotary  field  induc- 
tion motor  the  resistance  of  the  secondary  circuit  is  usually 
constant  when  the  motor  is  in  normal  operation,  but  the  fre- 
quency and  voltage  vary  together,  which  results  in  an  effect 
equivalent  to  varying  the  resistance  as  in  the  case  of  a trans- 
former, f Therefore,  the  induction  motor  diagram  is  similar 
in  principle  to  that  of  Fig.  277  for  the  transformer,  but  in  the 
motor  diagram  the  diameter  of  the  current  locus  is  relatively 
much  shorter,  on  account  of  the  greater  magnetic  leakage 
caused  by  the  air  space ; and  the  motor  current  varies  less  in  the 
windings  of  small-sized  motors  from  standstill,  which  is  equiva- 
lent to  the  condition  of  secondary  terminals  short-circuited  in 
the  transformer,  to  nearly  synchronism,  which  is  equivalent  to 
secondary  circuit  open  in  the  transformer,  than  is  the  case  for 
transformers.  Figure  463  shows  a diagram  for  an  induction 
motor  which  is  similar  to  the  transformer  diagram  of  Fig.  277, 
in  which  as  before  the  ratio  of  transformation  is  for  convenience 
taken  equal  to  unity,  and  reference  should  be  made  to  the 
method  of  construction  of  that  figure.  J The  exciting  current 
SO  is  taken  to  be  constant  though  it  varies  somewhat  more  in 
both  length  and  angle  than  in  a transformer  because  of  the 
greater  portion  of  the  circumference  of  the  current  locus  which 
comes  within  the  limits  of  no  load  and  full  load.  The  primary 
power  factor  is  a maximum  when  the  primary  current  vector  FS, 
closing  the  triangle  of  which  the  other  two  sides  are  the  exciting 
* Art.  70  (a),  Case  1.  t Art.  70  (&),  Case  5.  \ Art.  129. 


804 


ALTERNATING  CURRENTS 


current  SO  and  tlie  secondary  current  FO,  is  tangent  at  F to  the 
secondary  current  locus  OFMX.  For  smaller  primary  currents 
the  exciting  current  component  SO  causes  the  lag  angle  between 
the  primary  current  and  the  vertical  impressed  voltage  line  OE 
to  be  larger  than  at  F , the  point  of  tangency,  and  for  larger 


currents  than  FS  the  magnetic  leakage  of  the  primary  and 
secondary  circuits  causes  the  angle  to  be  greater  than  at  tan- 
gency. At  F"  on  OFMX  is  shown  the  current  locus  point  of 
maximum  power  input,  as  at  this  point  the  projection  of  the 
primary  current  on  the  voltage  vector  OE  has  its  maximum 
value.  The  primary  current  at  no  load  is  SO  in  the  diagram; 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


805 


the  exciting  component,  which  may  be  considered  constant 
without  serious  error,  and  the  component  due  to  the  secondary 
current  are  at  that  time  of  very  small  value,  the  latter  hav- 
ing an  active  component  just  sufficient  to  drive  the  armature 
against  bearing  friction  and  windage  and  to  supply  magnetiza- 
tion losses.  The  component  of  the  primary  current  required 
to  oppose  the  magneto-motive  force  of  the  no  load  secondary 
current  is  added  vectorially  to  the  true  primary  exciting  current 
and  is  considered  part  thereof.  The  standstill  or  starting  cur- 
rent, when  starting  resistance  or  other  current-reducing  devices 
are  not  used,  is  OF'".  This  is  located  on  OFMX  at  the  point 
where  the  active  voltage  OQ'"  shown  in  the  voltage  locus  con- 
struction is  all  absorbed  in  the  resistance  of  the  primary  and 
secondary  windings.  At  this  point  the  drop  of  impressed 
voltage  due  to  magnetic  leakage  is  Q'"F.  This  is  the  same  as 
the  short-circuit  point  in  a transformer  diagram.*  The  power 
input,  output,  and  all  conditions  of  working  may  be  obtained 
from  this  locus,  as  has  already  been  explained  for  the  trans- 
former,-)- or  a different  method  described  fully  by  McAllister, 
which  is  sometimes  more  convenient,  may  be  used  for  the  pur- 
pose. J Using  Fig.  463,  the  triangle  of  primary  voltages  OEQ' , 
for  any  secondary  current  as  OF',  is  similar  by  geometrical  con- 
struction to  OF'c , where  F'c  is  a line  drawn  from  F'  to  c at  right 
angles  to  OX.  The  side  OQ'  of  OFQ'  is  proportional  to  F'c. 
The  points  a and  b are  laid  off  on  F'c  so  that  F'a  is  proportional 
to  0 K,  the  impressed  voltage  drop  due  to  the  torque  of  the  motor 
load  ; ab  is  proportional  to  KJ,  the  drop  due  to  the  secondary 
resistance ; and  be  is  proportional  to  J Q' , the  drop  due  to  the 
primary  resistance.  The  distance  cd  is  equal  to  the  active  com- 
ponent of  the  no  load  current.  Then,  since  F'd  is  the  active 
component  of  the  primary  current,  it  is  proportional  to  the 
power  input ; also,  F'a  is  proportional  to  the  power  output ; 
ab  is  proportional  to  the  I2R  loss  in  secondary  resistance  ; be  is 
proportional  to  the  I2R  loss  in  primary  resistance ; and  cd  is 
proportional  to  the  no  load  power  loss.  The  lengths  of  these 
lines  respectively  multiplied  by  a value  representing  the  im- 
pressed voltage  OF  (that  is,  taken  to  a proper  scale)  are  there- 

* Art.  129.  t Art.  130. 

X McAllister,  Sibley  Journal  of  Engineering , Nov.,  1904,  and  Alternating 
Current  Motors,  3d  Ed.,  p.  105  et  seq. 


806 


ALTERNATING  CURRENTS 


fore  equal  to  the  motor  input  and  output  and  the  several  losses. 
The  angle  Q'KE  is  considered  constant  for  all  positions  of  Q\ 
hence  the  angle  Oac  is  constant  for  all  values  of  F,  and  the  two 
straight  lines  drawn  from  0 to  F'"  and  0 to  M,  through  a and 
l respectively,  divide  every  vertical  line  dropped  from  any 
point  on  the  current  locus  to  the  line  Sd  extended  into  seg- 
ments proportional  to  the  power  quantities. 

F'  a 

From  the  construction  it  is  seen  that  equals  the  efficiency 

when  the  secondary  current  is  OF' . The  corresponding  ratio 
decreases  to  a value  of  zero  when  the  motor  is  standing  still  or 
running  at  no  load,  and  increases  to  a maximum  when  F is  at  a 
point  on  the  locus  where  the  tangent  to  the  locus  is  parallel  to 
OF'".  The  length  F'b  is  proportional  to  the  torque  measured 
in  terms  of  watts  received  by  the  secondary  circuit  divided  by 
the  theoretical  synchronous  speed,  and  it  grows  from  a value  of 
zero  when  the  secondary  current  is  zero  to  a maximum  value 
when  F is  at  a point  on  the  locus  where  the  tangent  is  parallel 

to  OM.  The  speed  is  proportional  to  , since  the  speed  equals 


the  output  divided  by  the  torque.  The  slip  for  varying  loads  is 

proportional  to  since,  as  shown  by  the  formulas,*  the  slip  is 

F b 

proportional  to  the  secondary  copper  loss  divided  by  the  torque, 
provided  K of  the  formulas  remains  constant. 

When  the  impressed  primary  voltage,  the  no  load  primary 
current,  the  no  load  primary  angle  of  lag,  the  resistances  and 
standstill  reactances  of  the  primary  and  secondary  windings, 
and  the  synchronous  speed  of  an  induction  motor  are  known,  it 
is  possible  to  construct  accurate  curves  of  its  performance  from 
no  load  to  full  load.  In  fact,  test  curves  of  motor  performance 
closely  approximate  those  that  are  derived  from  the  circle  dia- 
gram. In  using  the  diagram  it  should  be  noted  that  the  ex- 
citing current  is  assumed  to  be  the  same  in  all  the  phases,  and 
that  all  currents  represented  are  for  a single  phase,  so  that  the 
power  and  losses  obtained  from  the  products  of  current  and  vol- 
tage vectors  must  be  multiplied  by  m , where  m is  the  number  of 
phases.  The  exciting  current  line  S 0 should  have  an  active  com- 
ponent equal  to  the  no  load  losses  divided  by  m and  a quadra- 


* Art,  190. 


ASYNCHRONOUS  MOTORS  ANI)  GENERATORS 


807 


1 2 

ture  component  equal  to , . x — times  the  magneto- 

ni  X V2  X 1.25  m 

motive  force  necessary  to  create  a flux  equal  to  that  of  the  rotat- 
ing field,  as  shown  in  Art.  188. 

Prob.  1.  Construct  the  circle  diagram  and  from  that  draw 
the  curves  — from  no  load  speed  to  standstill  — of  efficiency, 
power  factor,  torque,  power  input,  copper  loss,  slip,  speed,  and 
primary  and  secondary  currents  (considering  the  ratio  of  trans- 
formation unity)  as  functions  of  the  power  output  used  as  the 
abscissas,  for  a rotary  field  two-phase  induction  motor  having 
the  following  characteristics  : full  load  capacity  rating  25  horse 
power  ; pairs  of  poles  6 ; connection  quarter-phase  ; impressed 
volts  per  phase  440  between  loads ; no  load  losses  2 kilowatts, 
which  are  assumed  fixed  for  all  loads.  The  no  load  current  is 
15  amperes  per  phase  and  the  no  load  power  factor  is  20  per  cent. 
The  standstill,  or  short-circuit  current,  is  125  amperes,  and  the 
standstill  power  factor  is  50  per  cent.  The  secondary  winding  is 
also  assumed  to  be  quarter-phase  and  the  resistances  and  stand- 
still reactances  of  the  primary  and  secondary  windings  to  be  equal. 
Figure  463  indicates  the  process.  The  circle  diagram  may  be 
drawn  and  the  currents  laid  out  for  a single  phase,  as  suggested 
above,  using  one-half  of  the  total  power  as  the  full  load  output. 
The  standstill  reactance  for  either  the  primary  or  secondary 
winding  may  be  obtained  by  dividing  one  half  the  product  of 
the  voltage  and  standstill  induction  factor  (sin  0)  by  the  stand- 
still current.  The  resistance  of  either  winding  is  equal  to  one- 
half  of  the  product  of  the  voltage  times  the  standstill  power 
factor  (cos  0 ) divided  by  the  current.  The  diameter  of  the 
E 

locus  is  D = , where  X , and  X0  are  the  standstill  re- 

Xi+X* 

actances  of  the  primary  and  secondary  windings.  The  stand- 
still current  is  very  large,  so  that  some  error  is  introduced  in 
finding  the  diameter  by  using  the  reactances  as  here  deter- 
mined. More  accurate  methods  will  be  given  later. 

192.  Substituted  Impedance  for  the  Rotary  Field  Induction 
Motor.  — From  the  circle  diagram  and  the  formulas  of  the  pre- 
ceding article  it  is  evident  that  equivalent  impedance  can  be 
substituted  for  the  rotary  field  induction  motor,  as  was  done  in 
the  case  of  the  transformer.  In  this  case  the  load  must  be 
represented  by  non-reactive  resistance  having  a value  equal 


808 


ALTERNATING  CURRENTS 


to  the  watts  output  per  phase  divided  by  the  square  of  the  sec- 
ondary current  reduced  to  the  primary  equivalent.  This  latter 
is  equal  to  the  primary  current  per  phase  after  the  no  load  cur- 
rent per  phase  has  been  vectorially  deducted  from  it.  Figure 
464  shows  the  arrangement.  In  this  figure  R,  and  Xf  repre- 
sent an  impedance  which  will  permit  the  exciting  current  to 
flow;  Rx  and  Xx  are  the  resistance  and  reactance  of  the  primary 
coils  ; R2  is  the  resistance  of  the  secondary  coils  ; X2  is  the  re- 
actance of  the  secondary  coils  at  standstill  ; s2  is  the  slip ; and 
R is  the  resistance  that  would  be  required  when  the  slip  is  s2 
to  dissipate  power  equal  to  the  mechanical  load  of  the  motor 
when  connected  into  the  circuit  as  indicated  in  the  figure.  The 


a c e 


Fig.  4G4.  — Arrangements  of  Impedance  which  are  Approximately  Equivalent  to  the 
Circuits  and  Load  of  a Rotary  Field  Induction  Motor. 

impedances  of  the  primary  and  secondary  windings  should  be 
the  impedances  of  one  phase  for  each,  and  the  secondary  quan- 
tities should  be  properly  reduced  to  primary  equivalents.  The 
coils  arranged  to  absorb  the  exciting  or  no  load  current  should 
absorb  a current  comprising  a quadrature  component  equal  to 

the  — — — times  the  magneto-motive  force  of  the  coils  of  a 

V2  x 1.25  nx 

single  phase,  combined  vectorially  with  an  active  component  rep- 
resenting — times  the  total  no  load  losses,  where  m equals  the 
m 

number  of  phases;  this  is  of  course  equal  to  the  no  load  current 
measured  in  one  phase. 

The  solution  of  these  circuits  by  impedance  and  conductivity 
formulas  is  the  same  as  in  a simple  transformer,*  except  that 


* Art.  133. 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


809 


the  total  motor  output  and  losses  are  equal  to  the  values  given 
by  the  formulas  multiplied  by  the  number  of  phases.  The  no 
load  current  varies  in  value  and  phase  position  more  than  in 
the  case  of  a well-designed  constant  potential  transformer,  so 
that  it  is  necessary,  where  minute  accuracy  is  for  any  reason 
demanded,  to  transfer  the  no  load  circuit  (Fig.  464)  from  ab 
to  cd. 

The  secondary  induced  voltage  is  all  absorbed  in  the  imped- 
ance of  the  secondary  windings.  In  terms  of  Fig.  464  in  which 
the  secondary  quantities  are  all  reduced  to  terms  of  the  primary 
circuit,  it  is  the  voltage  measured  from  c to  e.  It  varies  directly 
with  the  slip  and  has  a value  at  any  slip  s2  which  is  s2Ev  in 
which  E2  is  the  induced  voltage  corresponding  to  a frequency 
of  f periods  per  second  for  the  cycles  of  inducing  magnetism 
and  is  the  voltage  measured  from  c to  d in  Fig.  464.  The  in- 
duced voltage  in  the  primary  winding  is  also  E2.  From  Fig. 
464  it  will  be  observed  that 


in  which  Ix  is  the  total  current  in  the  primary  winding,  1^  is 
the  component  in  opposition  to  the  induced  secondary  current, 
If  is  the  exciting  current,  Ex  is  the  primary  impressed  voltage, 
Z is  the  impedance  of  the  motor  circuit  from  a to  b through  c*, 
e , /,  and  d,  and  Zs  is  the  impedance  from  a to  b through  the 
parallel  path. 

Also,  obviously, 


S2E2  — E2  (^2  4*  JS 2^2) 

and  Ex  — E2+  I2  (Rt  +jX1)  = /2  (ri  + - +j  (A)  4-  A2)  • 


j 


From  which, 


s2E2  = h (E2  + s22-Y22)’’ 


and  SgAj  = J2  [(*2#!  + R2Y  + s22(A1  + X2)2]J. 


Therefore 


810 


ALTERNATING  CURRENTS 


Also,  the  torque  per  phase  is  equal  to 
T = K'<t>n2I2  cos  02  ; 

108_EL 


but 


V2ttAw2/ 


s2^2 


and 


cos  d2  _ 


W + S,2^ 

7L 


Therefore, 

T = K •S2^22^2  _ X 

R*  + s22X22  ( 


s2EfR2 


h R,  + ^2)2  + «22(^l  + ^2)2’ 


in  which  K is  a numerical  constant. 

The  torque  is  therefore  a function  of  the  slip.  If  the  resist- 
ance of  the  armature  circuit  of  a motor  is  varied,  while  Ev  Rv 
Xx  and  X2  are  maintained  unchanged,  the  torque  will  pass 
through  a maximum  value  when  dT/ds2  = 0,  which  value  is 

y 2 

T =1 K — 1 

max  2 R1  + [R*  + (X1  + X^f 


This  equation  shows  that  the  maximum  torque  of  any  induc- 
tion motor  is  proportional  to  the  square  of  the  impressed  vol- 
tage, and  that  its  value  is  independent  of  R2.  The  primary 
current  producing  it  is  also  independent  of  R2 ; but  the  slip  at 
which  it  occurs, 


R, 

[^+(V1  + X2)2]*’ 


is  directl}r  proportional  to  R2. 

If  the  armature  core  losses  are  relatively  large  and  vary  with 
the  slip,  as  would  be  the  case  if  the  armature  core  were  made 
of  unlaminated  hard  cast  iron,  the  apparent  resistance  of  the 
armature  winding  might  be  mostly  the  effect  of  core  losses, 
which  vary  with  the  slip,  and  the  motor  might  then  produce  a 
uniform  torque  (like  a series-wound,  direct-current  motor  sup- 
plied with  a constant  current)  over  a considerable  range  of 
speeds. 

The  resistance  of  the  armature  conductors  is  usually  such 
that  the  maximum  torque  comes  at  from  1 to  10  per  cent  slip, 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


811 


or  s2  equals  from  .01  to  .1 ; and  in  practice  an  external  resist- 
ance is  usually  introduced  into  the  armature  windings  at  start- 
ing,* which  serves  both  to  increase  the  torque  at  starting  and 
to  avoid  the  excessive  rush  of  current  which  might  otherwise 
occur  while  the  armature  is  stationary.  Figure  465  shows  the 
relation  of  torque  to  slip  for  a motor  when  the  armature  circuits 
have  resistances  of  .02,  .045,  .18,  and  .75  ohm.f  This  shows 
plainly  that  the  torque  can  be  caused  to  have  a maximum  value 
at  different  slips  from  100  per  cent  to  10  per  cent  of  the  field 


Fig.  4(i5. — Torque  Curves  for  a Rotary  Field  Motor  with  Different  Resistances  intro- 
duced into  the  Secondary  Winding. 

frequency  by  gradually  reducing  the  resistance  of  the  armature 
circuit  from  .18  to  .02  ohm  as  the  speed  of  the  armature  in- 
creases. An  increase  of  armature  resistance  above  .18  ohm 
brings  the  torque  to  a maximum  value  when  the  armature  is 
driven  backwards  by  external  mechanical  power,  thereby  in- 
creasing the  slip  beyond  100  per  cent. 

Induction  motors  are  usually  designed  to  run  normally  at  a 
speed  which  is  between  synchronism  and  the  speed  giving  the 
greatest  torque.  In  designing  them  X2  is  made  as  small  as 
practicable,  by  reducing  the  magnetic  leakage  to  a minimum, 
and  R2  is  then  given  such  a value  that  the  slip  at  normal  full 

t Steiumetz,  Trans.  Amer.  Inst.  E.  E.,  Vol.  XI,  p.  700. 


* Art.  196. 


812 


ALTERNATING  CURRENTS 


load  is  sufficient  to  give  a value  of  the  torque  which  is  from 
one  fourth  to  three  fourths  of  its  maximum  value.  Such 
motors  can  therefore  carry  considerable  overloads,  as  they 
normally  operate  at  speeds  between  synchronism  and  the  speed 
of  maximum  torque  ; but  if  the  resisting  moment  of  the  load  is 
increased  beyond  the  maximum  torque,  the  motor  stops.  In 
this  respect,  induction  motors  differ  from  shunt- wound  direct- 
current  motors  operated  at  constant  voltage,  in  which  the 
torque  increases  in  direct  proportion  with  the  armature  cur- 
rent and  therefore  with  the  resisting  moment  of  the  load, 
provided  the  total  magnetic  flux  passing  through  the  armature 
remains  constant  and  does  not  shift.  Increasing  the  load  on 
a well-designed  shunt-wound  direct-current  motor  will  not 
ordinarily  stop  it  until  the  armature  windings  are  melted,  or  in 
small  motors  until  the  drop  of  voltage  due  to  current  flowing 
through  the  armature  conductors  is  equal  to  the  impressed 
voltage. 

Near  synchronism  the  leakage  reactance  of  induction  motors 
becomes  negligible,  and  from  the  formula  it  is  seen  that  the 
torque  varies  almost  directly  as  the  slip.  This  is  approximately 
true  through  the  ordinary  range  of  slips  from  no  load  to  full 
load  in  commercial  induction  motors. 

The  torque  of  the  armature  when  the  speed  is  V revolutions 
per  minute  and  the  output  P is  given  in  watts  is  equal  to 

T = — — in  dyne-centimeters, 


mn 


P X 107 
2 rrV  16.3 


in  gram-centimeters, 


r"  = 2^226  hl  P°UDd'feet- 

Since  the  power  equals  2 7t  times  the  product  of  torque  and 
speed,  and  it  has  already  been  shown  that 

s0E„Ro 


p = 

60  Z22 


therefore, 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


813 


V V 
in  which  S2  = — r and  s2  = — , and  consequently,  since 

V V 


o V V-  V 

s*=v=~r 


1 V H 
1 1 Sg, 


P — J_  g2  ) /i*.2^2 

60  K2  + s22X22  ’ * 


which  readies  a maximum  value  at  a lesser  slip  than  the  torque. 

The  sum  of  the  output  of  the  motor  plus  the  armature 
core  losses  and  friction  is  equal  to  the  continued  product  of 
(1)  the  number  of  armature  conductors,  (2)  the  effective 
motor  load  current  in  each  conductor  for  the  given  output, 
(3)  the  voltage  which  would  be  developed  respectively  in 
each  conductor  if  the  armature  were  driven  at  its  running  speed 
in  an  equal  stationary  field,  and  (4)  the  cosine  of  the  angle  of  lag 
of  the  armature  current  with  respect  to  the  induced  voltage,  or, 


P — W 2 ^2r'^2  ^2c  C0S  ^2 c’ 
but 

J _ S2-^2c . 

2i 

where  nv  Lc , s2E2c,  ^20  X2 c,  and  d2c  are  the  number  of  con- 
ductors and  the  current,  voltage,  resistance,  reactance,  and 
angle  of  lag  for  any  conductor,  the  subscript  c indicating  that 
the  quantities  are  for  a single  conductor ; and 


cos  92c  — 


Bo 


Therefore, 

also 


vv+w 


2c 


P _ . 


-^2  c — K — 


1 


where  n1  is  the  number  of  primary  conductors,  and  therefore 


1 F2>S.2s2B, 


2c 


E2S2s2R2c 


P = 


V + S2X2c  ft2(i?2c2  + S2X2c) 


This  formula  is  not  one  which  is  ordinarily  needed  in  the 
design  of  a motor,  but  it  plainly  shows  the  effect  on  the  output 


814 


ALTERNATING  CURRENTS 


of  a motor  which  is  caused  by  varying  any  one  of  its  construc- 
tive details  while  the  others  remain  unchanged.  A very  im- 
portant deduction  from  the  formula  is  that  the  torque  and 
output  of  an  induction  motor  vary  as  the  square  of  the  primary 
voltage,  so  that  a machine  which  will  carry  an  overload  of  50  per 
cent  on  its  normal  voltage  will  barely  run  at  full  load  if  the  vol- 
tage is  reduced  20  per  cent.  The  formula  also  shows  that  the 
slip  is  inversely  dependent  on  the  square  of  the  primary  voltage. 

The  motor  torque  is  shown  from  the  equivalent  impedance 
circuits  to  be  approximately,  in  gram-centimeters, 

rp_  HI?  107 

2ttV  16.3’ 


where  V is  the  revolutions  of  the  motor  secondary  per  minute 
and  Rl 22  is  the  motor  rotative  output.  The  slip  is 

, =_A_. 

2 R2  + R 

The  number  of  revolutions  per  minute  the  motor  armature 
speed  is  less  than  that  of  the  rotating  field  is 

v — V ^2 = PPX ^2 

R + R2  p R + R 2 ' 

since  the  speed  of  the  rotating  field  in  revolutions  per  minutes. 

V,  is  equal  to  PPX,  where  /is  the  frequency  of  the  primary 
P 

circuit  and  p is  the  number  of  pairs  of  motor  poles.  Likewise, 
the  actual  speed  of  the  armature  in  revolutions  per  minute  is 


V = V(  1 — s ) = ^ ■ 

2 P r + r2 


Methods  of  determining  other  characteristics  are  apparent  from 
the  study  of  the  induction  motor  formulas  and  Figs.  464  and 
465.  The  processes  used  in  dealing  with  substituted  impedances 
for  the  induction  motor  are  very  similar  to  those  required  for 
the  transformer.  It  will  be  remembered  that  to  reduce  sec- 
ondary resistances  and  reactances  to  primary  equivalents,  they 
must  be  multiplied  by  s2  where  s is  the  ratio  of  transformation. 

l3rob.  1.  Determine  the  substituted  impedances  for  the  motor 
given  in  the  Problem  at  the  end  of  Art.  191. 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


815 


Prob.  2.  From  the  substituted  impedances  obtain  the  pri- 
mary current  and  power  factor  for  full  load,  half  load,  and 
quarter  load  for  the  motor  of  Prob.  1. 

Prob.  3.  From  the  substituted  impedances  of  Prob.  1 find 
the  torque  and  slip  at  full  load,  half  load,  and  quarter  load. 

193.  Additional  Formulas  for  Torque  and  Slip  of  a Polyphase 
Induction  Motor.  Circle  Diagram  of  Magnetic  Fluxes. — In  the 
last  article,  the  slip  was  expressed  thus, 

s = ^2  . 

2 R2+R 

Multiplying  this  by  the  secondary  current  I2,  squared,  there 
results 

I2R0 

82  i22r2  + 12R' 

where  the  numerator  represents  the  armature  copper  losses 
and  the  denominator  represents  the  same  copper  losses  plus  the 
power  output  (72  should  be  of  such  value  that  this  will  in- 
clude frictional  losses)  ; thei-efore, 


where  Pc  equals  the  armature  copper  losses  and  equals  the 
power  transformed  from  the  primary  circuit  into  the  secondary 
circuit.  The  iron  losses  may  be  considered  as  all  absorbed 
from  the  primary  circuit. 

The  value  of  torque  stated  in  pound-feet  is 


T= 


Pn  10* 


2 77-  V ■ 226’ 

where  Pa  is  the  output. 

P0  = 7272,  where  R is  the  equivalent  secondary  substituted 
load  impedance  and  Pt  = PR  + Z2722.  Therefore, 

R 


P = P 

n 1 


R + 72/ 


but  V'=V- 


R 


60/  R 


R 4-  R2  p R + R2 
P0  and  V in  the  formula  for  torque  gives 

- 117  £3 

/ 2 7T  • 226  • 60  ' / 


Substituting  these  values  of 


T=  P- 


816 


ALTERNATING  CURRENTS 


The  secondary  current  of  the  motor  when  the  armature  has 
no  external  load,  expressed  in  primary  equivalents,  may  be  ob- 
tained with  approximate  accuracy  by  vectorially  subtracting  the 
primary  current  at  no  load  from  the  primary  current  which  flows 
at  any  input  under  consideration.  Likewise,  the  resistance  of 
a closed  circuit  secondary  winding  can  be  calculated  by  observ- 
ing the  primary  power  input  and  current  when  a low  voltage  is 
applied  to  the  primary  windings  and  the  armature  is  locked  in 
a stationary  position.  The  difference  between  the  calculated 
primary  I2R  loss  and  the  observed  power  is  approximately  the 
secondary  I2R  loss.  Dividing  this  difference  by  the  square 
of  the  observed  current  gives  the  approximate  armature  resist- 
ance in  primary  equivalents.  The  current  used  should  be  less 
than  that  for  full  load  in  order  that  the  magnetization  losses 
may  be  negligible.  However,  correction  can  be  made  as  in  a 
transformer.* 

In  an  earlier  article  it  was  stated  that  the  leakage  reactance  of 
a polyphase  motor  decreases  on  account  of  the  saturation  of  the 
core  teeth  with  the  heavy  currents  of  short  circuit.  In  order 
to  obtain  a value  of  Xx  + X2  to  be  used  in  laying  out  the 
diameter  of  the  circular  locus  of  currents,  it  is  well  to  experi- 
mentally determine  the  value  when  the  current  is  at  or  near  its 
value  for  full  load.  This  can  be  done  by  locking  the  armature 
in  a stationary  position  and  applying  a sufficiently  reduced  vol- 
tage to  the  primary  winding  so  as  to  cause  the  desired  current  to 
flow.  Then,  if  the  observed  voltage,  current,  and  watts  are  E' , 


Ir,  and  P\  we  have 


O'  = cos-1 


ET 


where  O'  is  the  angle  of  lag.  From  which  the  reactive  com- 
ponent of  the  voltage  is 

E'  sin  O'  = E'x  = I'X'  and  X'  = — , 

where  E' x is  the  component  of  the  voltage  produced  by  the 
change  of  the  leakage  flux,  and  X'  = Xx  + X2  is  the  combined 
leakage  reactance  of  the  primary  and  secondary  circuits.  Then 
the  diameter  of  the  current  locus  for  a machine  having  leakage 
reactance  X'  when  current  F flows  is 


I)  - 


ET 

E'  sin  O'  ’ 


* Art.  147. 


ASYNCHRONOUS  MOTORS  AND  GENERATORS  817 

where  E is  the  normal  motor  voltage;  or,  if  the  assumption  is 
made  that  the  reactance  of  the  primary  and  secondary  circuits 
are  equal,  — which  will  not  ordinarily  introduce  a very  large 
error,  — and  we  know  the  current,  voltage,  watts,  and  slip,  and 
hence  6 when  the  machine  is  running  at  the  working  load  de- 
sired, we  can  calculate  the  combined  leakage  reactance  at  that 
load.  Thus, 

E sin  d XT-  v~\ 

■ — — — — (at  + s2xy, 

where  X is  the  leakage  reactance  of  either  winding  ; hence, 
where  s2  is  the  slip 

_ E sin  6 
7(1  + s2) 

and  the  diameter  of  the  current  locus  circle  is 

EI(1  4-  s2) 7(1  + s2) 

2 E sin  6 2 sin  9 

In  laying  out  the  circle  diagram,  it  is  possible  to  consider 
the  diameter  of  the  locus  extended  to  the  origin  of  the  primary 
current  vector  as  being  proportional  to  a magnetic  flux,  if  the 
power  component  of  the  exciting  current  is  neglected.  Thus, 
consider  OFCA,  Fig.  466,  the  parallelogram  of  currents  in  a 
motor  when  the  primary  current  is  OF  = Iv  the  secondary  cur- 
rent is  OA  = I2,  and  the  exciting  current  is  00=1^  (neglect- 
ing the  power  component  of  the  exciting  current  for  the  present). 
The  impressed  voltage  is  0EV  This  is  the  phase  diagram  of 
currents  for  a transformer  with  magnetic  leakage  * which  is 
similar  in  voltage  and  current  relations  to  a polyphase  induc- 
tion motor.  For  convenience,  consider  that  the  primary  current 
sets  up  a magneto-motive  force  which  sets  up  the  mutual  flux 
through  the  primary  and  secondary  windings,  the  leakage  flux 
about  its  own  windings,  and  neutralizes  any  tendency  of  the 
secondary  current  to  set  up  leakage  flux.  The  secondary  cur- 
rent and  the  component  of  voltage  produced  by  the  mutual  flux 
thus  considered  are  in  the  same  phase,  while  the  leakage  flux 
about  the  primary  conductors  is  due  to  the  magnetic  paths  not 
only  across  the  teeth  in  the  field  core,  but  also  across  the  teeth 
of  the  armature  core.  This  assumption  is  not  necessary,  but  it 

* Art.  124. 

3g 


818 


ALTERNATING  CURRENTS 


simplifies  the  conception  without  introducing  error.  Draw  OB 
upon  00  extended  so  that 

OB  = 4 = c p 

10  R 1 

where  nl  are  the  primary  turns,  <3?,  the  total  flux,  and  R the 
reluctance  of  the  magnetic  leakage  and  mutual  magnetic  paths 
in  parallel,  or  where 

B MRXLRiL 

[iMR\L  + RMR,L  + BXLR2L 

where  RM  is  the  reluctance  of  the  path  of  flux  which  passes 
through  both  windings,  RXL  is  the  reluctance  of  the  path  for 


Fig.  466.  — Circle  Diagram  for  showing  Relation  of  Fluxes,  Slip  Line,  and  Circle 
Loci  for  determining  other  Characteristics  of  a Polyphase  Induction  Motor. 


primary  leakage  flux  which  passes  across  the  teeth  of  the  field 
core,  and  R2L  is  the  reluctance  of  the  path  for  leakage  flux 
around  the  primary  windings  which  passes  across  the  teeth  of 
the  armature  core  and  around  the  air  gap.  OB  then  equals  the 
total  flux  set  up  in  the  magnetic  circuits  of  the  machine,  and 
it  is  produced  by  the  magnetizing  current  00.  The  leakage 
component  of  this  flux  must  be  in  phase  with  the  total  primary 
current,  if  the  iron  loss  angle  is  neglected  ; and  by  properly 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


819 


adjusting  the  flux  scale  in  which  OD  and  other  fluxes  are  laid  out 
it  may  be  represented  in  phase  and  magnitude  by  OF , where 

OF=  ^FILiLx  = q> 

10  Rl 

where  <E>Z  is  the  leakage  flux  and  RL  is  the  combined  reluctance 
of  the  leakage  paths  in  parallel.  The  mutual  flux  must  be  the 
vector  difference  of  the  two  noted,  or 

®M=  ®t~®L=FI). 


But  the  mutual  flux  <J>  is  cut  by  the  secondary  conductors  and 
sets  up  the  secondary  voltage  Ev  which  must  be  at  right  angles 
to  it.  As  the  secondary  voltage  is,  by  the  assumption,  in  phase 
with  the  secondary  current  Iv  the  latter  also  must  be  in  quad- 
rature with  T >M . Therefore,  as  FC  in  Fig.  466  equals  OA , the 
angle  DFO  is  always  a right  angle.  Therefore,  as  F moves, 
due  to  changes  in  the  load  and  hence  the  primary  and  secondary 
currents,  it  must  describe  an  arc,  OFMD.  If  there  were  no 
losses  this  locus  would  be  a complete  semicircle. 

The  output  when  the  secondary  neutralizing  component  of 
the  primary  current  is  OF  is  evidently  Fl  x FP,  or  at  proper 
scale  equals  FP  if  there  are  no  losses.  A primary  resistance 
loss  will  reduce  this  because  the  induced  primary  and  secondary 
voltages  will  be  smaller  by  the  effect  of  the  drop  in  the 

primary  windings.  This  means  that  FP,  the  mutual  flux,  must 
be  shorter  in  equal  proportion.  In  the  same  way  the  secondary 
output  must  be  less  by  reason  of  the  resistance  drop  in  the  sec- 
ondary windings,  and  this  may  also  be  considered  as  equivalent 
to  the  shortening  of  the  mutual  flux  line.  Suppose,  now,  the 
motor  is  loaded  until  it  comes  to  a standstill,  and  that  the  cur- 
rents have  increased  until  IP  has  moved  to  F' . Since  the  arma- 
ture is  doing  no  work,  the  secondary  voltage  set  up  by  the 
mutual  flux  FD  is  now  all  expended  in  driving  the  current  F'  C 
through  the  secondary  windings.  The  proportion  of  the  in- 
ductive effect  of  the  mutual  flux  used  in  overcoming  IR  drop 


per  ampere  is  then  measured  by  the  ratio 


F'D 

OF 


if  the  copper 


loss  of  the  exciting  current  00  is  neglected  and  the  currents 
are  reduced  for  the  ratio  of  transformation  of  unity  as  shown 
in  the  figure.  Since  the  same  relations  must  hold  for  all  cur- 


820 


ALTERNATING  CURRENTS 


rents  if  the  reluctances  of  the  magnetic  circuits  are  considered 
to  be  constant,  the  proportion  of  the  inductive  effect  of  the 
total  flux  which  is  absorbed  by  IR  drop  for  any  other  current, 
such  as  CF , is  FD'  where  the  triangle  CFD'  is  made  similar 
to  CF'  D.  But  F is  any  point  on  the  arc  CFF'  and  angle  FD'  C 
is  constant,  and  therefore  angle  CD'D  is  constant  for  any  posi- 
tion of  F on  the  locus  CMD.  Hence  the  point  D'  must  trace  the 
arc  of  a circle  CM" D,  having  the  chord  CD.  If  U is  the  center 
of  the  chord,  the  perpendicular  UN  lies  on  the  radius  of  the 
circle  CD'D , and,  D being  one  point  of  the  circumference,  the 
center  of  the  circle  is  the  intersection  of  a normal  to  F'D  at  D 
with  UN,  since  F'D  must  be  tangent  to  the  circle  at  Z>,  as  the 
inductive  effect  of  the  mutual  flux  represented  by  F'D  is  just 
used  up  in  furnishing  the  IR  voltages.  The  power  lost  in  the 
resistance  of  primary  and  secondary  windings  when  any  current 
CF  flows  is  then  FR.  If  now  the  ordinates  of  the  line  ZZ'  are 
equal  to  the  assumed  constant  waste  of  power  in  iron  losses, 
the  losses  in  the  primary  winding  by  reason  of  the  copper  loss 
caused  by  the  flow  of  the  magnetizing  current  OG , and  the 
mechanical  frictional  losses,  then  the  net  power  given  to  the 
shaft  by  the  armature  when  the  secondary  current  CF  flows  is 
represented  by  the  line  RS. 

The  relative  primary  and  secondary  copper  loss  can  be  found 
in  this  way  : divide  F'D,  the  standstill  mutual  flux  which  is  all 
used  in  generating  the  primary  and  secondary  IR  voltages,  into 
two  parts,  such  that  F'B  is  proportional  to  the  flux  used  in 
generating  the  primary  resistance  drop  and  BD  to  that  used 
in  generating  the  secondary  resistance  drop.  Then,  the  mutual 
flux  used  in  generating  the  IR  drop  for  any  other  current  such 
as  CF  must  be  divided  in  the  same  proportions,  and  hence,  FD'  is 
divided  into  FB'  and  B'D' . Now  a triangle  drawn  between  the 
points  C,  B , and  D is  similar  to  a triangle  drawn  between  the 
points  C,  B',  and  D',  since  triangles  CF' D and  CFD'  are  similar. 
Hence,  as  F varies  in  its  positions,  the  angle  made  by  lines 
connecting  C and  B' , and  B'  and  D must  be  equal  whatever 
may  be  the  position  of  F.  The  locus  of  B'  is  then  the  arc  of  a 
circle  upon  the  chord  CD.  The  center  His  the  intersection  of 
the  perpendicular  UN  with  the  normal  to  BD  erected  at  center 
point  Q.  This  is  evident,  since  B and  D both  lie  on  the  arc  and 
the  line  BD  is  therefore  a chord  of  the  arc  between  B and  D. 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


821 


For  any  primary  current  OF  = /,  supposing  the  scales  in  which 
the  lines  are  measured  to  be  properly  adjusted,  we  have  the  pri- 
mary input  equal  to  FP  ; the  secondary  input  nearly  equal  to  TS ; 
the  secondary  output  equal  to  RS ; the  primary  copper  loss  equal 
to  FT ; the  secondary  copper  loss  equal  to  TR ; and  the  excita- 
tion and  frictional  losses  (assumed  constant)  equal  to  SP. 

The  total  armature  torque  for  any  primary  current  OF  may  be 
scaled  as  approximately  equal  to  TS,  since  CFx  FB=I2x 
which  is  proportional  to  the  torque.  But,  if  the  perpen- 
dicular B'  W = TS  is  dropped  from  B',  the  similar  triangles 
B'  WB  and  CFB  give  the  proportion  OF : B'  W: : OB  : B'B,  or 
OF  x B'B  = B'W  x OB.  Therefore,  as  OB  is  fixed  in  length, 
B'  W = TS  is  proportional  to  the  torque  and  can  be  scaled  to 
read  torque  directly.  The  load  torque  equals  TS  minus  the 
frictional  and  windage  torque  of  the  armature  itself. 

The  slip  for  any  primary  current  may  be  found  as  follows:  the 


secondary  current  I2  ac  $>Msv  where  s2  is  the  slip,  or  s2  ac 


A 


ac 


OF 

B'B' 


But,  as  the  triangle  CFB'  is  similar  for  all  positions  of  F,  the 


slip  may  be  taken  as  approximately  proportional  to 


OB' 

B’B 


There- 


fore lay  out  the  angle  BKT  equal  to  OB'B , then  triangles  OB'B 

OB'  BK 

and  LKB  are  similar  and  — — — = — — - . As  KB  is  constant  in 

B'B  KB 

length,  the  slip  is  approximately  proportional  to  the  intercept 
LK  made  from  the  fixed  line  JK  by  the  line  FB,  wherever  F 
lies  on  the  locus. 

This  diagram,  in  so  far  as  the  current  locus  is  concerned,  is 
similar  to  that  of  Fig.  463,  and  the  various  quantities  may  be 
determined  in  the  same  way  in  either,  with  the  exception  that 
in  Fig.  466  the  fixed  losses  are  subtracted  from  the  measuring 
line  dropped  from  the  end  of  the  current  vector,  while  in  Fig. 
463  the  voltage  is  made  a little  larger  and  the  fixed  losses  are 
added  to  the  same  line.  The  circle  diagram  of  Fig.  466,  with 
the  approximate  method  of  finding  the  losses,  etc.,  was  discov- 
ered by  Heyland,  Behrend,  and  others  early  in  the  art  of  induc- 
tion motor  manufacture,  but  it  is  only  approximate  at  the  best 
and  in  many  instances  is  quite  inaccurate.  The  diagram  of 
Fig.  463  is  more  accurate  and  is  therefore  preferable  to  use. 
The  point  F'  in  both  Fig.  463  and  Fig.  466  shows  the  stand- 


ALTE RNATING  CUHIl ENTS 


• 822 

still  point ; the  locus  from  F'  to  C represents  the  machine  as  a 
motor  for  the  entire  range  of  speeds  from  standstill  to  theoretical 
synchronous  speed  ; and  the  locus  from  F'  to  D represents  the 
machine  as  a frequency  transformer,  driven  backwards  (with 
respect  to  the  rotating  magnetic  field)  from  a speed  of  standstill 
to  infinite  speed. 

Either  of  these  circle  diagrams  given  for  the  induction  motor 
is  lacking  in  absolute  accuracy,  since  both  the  phase  and  the 
scalar  value  of  the  induced  voltage  of  the  induction  motor 
change  when  the  current  changes,  and  this  changes  the  phase 
and  scalar  value  of  the  exciting  current  as  well  as  the  radius  and 
position  of  the  locus  circle.  Attention  has  already  been  called 
to  the  fact  that  the  reluctances  of  the  leakage  paths  vary  with 
different  currents.  Suffice  it  to  say,  however,  that  the  diagrams 
given  are  sufficiently  accurate  for  many  practical  purposes,  and 
if  they  are  constructed  upon  the  basis  of  the  magnetic  path 
reluctances  to  be  found  at  or  near  full  load,  they  will  ordinarily 
approximate  to  accuracy  from  no  load  to  well  over  full  load. 

194.  The  Rotary  Field  Induction  Motor  as  an  Asynchronous 
Generator. — If  a rotary  field  induction  motor  is  driven  above 
synchronous  speed  by  a source  of  mechanical  power  attached  to 
the  shaft,  while  it  is  electrically  connected  to  a conductor  system 
in  which  there  are  voltages  which  would  run  the  machine  as  a 
motor,  it  no  longer  acts  in  the  capacity  of  a motor,  but  the  arma- 
ture generates  electric  power  and  delivers  it  to  the  supply  lines. 
In  this  case  the  primary  circuit  of  the  machine  continues  to 
receive  its  quadrature  current  from  the  external  supply  system, 
and  the  rotating  magnetic  flux  is  the  same  as  if  the  machine 
was  operating  as  a motor.  But  the  relative  motion  between 
the  primary  and  secondary  circuits  having  been  reversed,  — the 
secondary  circuit  now  rotating  faster  than  the  rotating  flux,  — 
the  active  component  of  current  generated  in  the  windings  of 
the  armature  is  reversed,  which  results  in  the  opposing  compo- 
nent of  the  primary  current  being  reversed.  This  gives  the 
locus  shown  in  Fig.  467.  The  upper  half  of  the  locus  OTXT' 
is  the  induction  motor  and  frequency  transformer  locus  and  the 
lower  half  the  asynchronous  generator  locus.  When  sufficient 
power  is  applied  to  the  pulley  to  drive  the  machine  at  theoret- 
ical synchronism,  only  the  primary  circuit  exciting  current  SO 
flows.  When  the  armature  speed  is  raised  above  synchronism. 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


823 


current  flows  in  the  armature  windings,  inducing  an  equal  and 
opposite  component  of  current  in  the  primary  windings,  which 
passes  out  to  the  external  circuit  to  which  the  machine  is  at- 
tached. Since  the  magnetic  flux  has  not  been  reversed,  however, 
the  primary  circuit  of  the  machine  must  continue  to  draw  from 
the  external  circuit  a component  of  lagging  quadrature  current 


Fig.  467.  — Locus  of  Currents  in  Induction  Machine  when  Run  as  a Motor  and  as  an 
Asynchronous  Generator. 

sufficient  to  furnish  the  requisite  magneto-motive  force.  This 
is  equivalent  to  a flow  from  the  machine  of  a leading  quadrature 
current.  Such  a machine  will  evidently  work  in  parallel  witli 
synchronous  alternators  or  other  apparatus  which  will  furnish 
this  exciting  current,  but  it  will  not  work  when  connected  to 
circuits  containing  only  resistance  and  self-inductance.  The 
exciting  current  may  be  obtained  by  placing  condensers  of  suf- 
ficient capacity  across  the  circuit  or  by  obtaining  the  required 
leading  current  from  a synchronous  motor  connected  in  the 


824 


ALTERNATING  CURRENTS 


circuit.*  The  asynchronous  generator  function  of  the  polyphase 
induction  machine  is  sometimes  made  use  of  on  traction  systems 
in  which  the  cars  are  driven  by  such  motors.  In  this  case, 
when  the  motors  are  properly  connected  in  tandem, f they  can 
be  caused  to  pump  power  back  into  the  line  when  running 
down  grade  or  being  brought  to  a stop. 

When  an  induction  generator  receives  its  quadrature  exciting 
current  from  synchronous  generators  of  fixed  speed,  the  current 
delivered  from  the  asynchronous  generator  is  of  the  same  fre- 
quency as  that  of  the  synchronous  generators,  regardless  of  the 
speed  of  the  armature  of  the  asynchronous  machine,  but  the 
output  changes  with  the  speed  in  the  manner  shown  by  the 
locus  diagram.  However,  when  the  quadrature  exciting  cur- 
rent is  provided  by  a condenser  or  a synchronous  motor,  no 
synchronous  machine  of  fixed  speed  being  connected  with  the 
circuit,  the  frequency  of  the  voltage  and  current  delivered  by 
the  asynchronous  machine  varies  with  its  armature  speed.  The 
locus  diagram  of  Fig.  467  is  not  applicable  to  the  latter  case. 

If  a circuit  receiving  power  from  an  induction  generator  be- 
comes short-circuited,  the  generator  at  once  ceases  delivering 
power  on  account  of  the  fact  that  the  magnetic  flux  disappears 
when  the  impressed  voltage  disappears.  This  is  contrary  to  the  ac- 
tion of  synchronous  generators,  which  are  self-reliant  in  magnetic 
field  and  continue  to  deliver  power  to  the  short-circuited  lines. 

195.  Some  Features  of  Construction  of  Rotating  Field  Induc- 
tion Motors.  — The  secondary  windings  of  induction  motors  are 
wound  upon  laminated  drums  or  equivalent  rings,  which,  with 
the  windings,  usually  constitute  the  rotating  part  or  rotor. 
The  arrangement  of  the  windings  may  be  of  several  forms: 

1.  Squirrel-cage  Form , in  which  single  embedded  bar  con- 
ductors are  placed  on  the  armature  core  and  all  connected  to- 
gether at  each  end  by  a copper  ring,  thus  making  a conductor 
system  similar  in  form  to  the  revolving  cylinder  of  a squirrel 
cage  (Fig.  461).  The  conductors  are  insulated  from  the  core 

2.  Independent  Short-circuited  Coils.  — In  this  form  of  wind- 
ing the  secondary  conductors  are  of  insulated  wire  wound  in 
independent  short-circuited  coils,  or  of  insulated  bars  connected 
by  end  connectors  in  such  a way  as  to  make  independent  short- 
circuited  coils. 


* Art.  174. 


t Art.  196  (6). 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


825 


Fig.  468. — Elementary  Diagram 
of  a Tri-phase  Armature 
Winding  for  a Secondary 
Stator  arranged  to  surround 
a Primary  Rotor  having  Eight 
Resultant  Poles. 


3.  Independent  Coils  short-circuited  in  Common.  — Here  the 
coils  are  wound  as  in  the  preceding  form,  but  instead  of  being 
short-circuited  independently,  all  the  ends  are  brought  to  a 
common  point,  or  pair  of  points,  one  of  which  may  be  at  the 
front  end  and  the  other  at  the  back  end  of  the  armature. 

4.  Coil-wound  Armatures  connected  to  External  Devices.  — 
The  windings  of  paragraphs  2 and  3 
can  be  arranged  for  connection  to  ex- 
ternal terminals  or  collecting  rings,  so 
that  external  resistances  may  be  in- 
cluded in  series  for  varying  the  torque 
or  slip.  The  connection  may  be  so  ar- 
ranged that  the  windings  form  single 
or  any  polyphase  system  of  circuits. 

It  is  evident  that  the  pitch  of  the 
coils  of  the  second  and  third  forms  of 
drum  windings  must  be  equal  to  an 
odd  number  of  times  the  pitch  of  the 
rotating  field  poles,  in  order  that  the 
voltage  set  up  in  the  conductors  may 
be  additive,  and  coils  may  therefore  be  diametral  or  chordal  in 
machines  with  an  odd  number  of  pairs  of  poles,  but  cannot  be 
diametral  in  machines  with  an  even  number  of  pairs  of  poles. 
The  actual  number  of  coils  is  a matter  of  perfect  freedom,  pro- 
vided it  is  a multiple  of  two  or  three,  and 
the  connections  of  conductors  is  properly 
made  so  that  the  surface  of  the  armature 
core  may  be  uniformly  covered.  A tri- 
phase winding  for  a stationary  drum  ar- 
mature, which  is  intended  to  surround  a 
rotating  primary  structure  having  eight 
poles,  is  shown  diagrammatically  in  Fig. 
Fig.  469.  — Elementary  468,  and  a three-coil  armature  which  is  in- 

Diagram  of  a Wound  ^enqeq  t0  revolve  within  a six-pole  primary 
Secondary  Rotor  to  be  . * 

structure  is  shown  in  t lg.  4b9. 

As  said  earlier,  it  is  a matter  of  indiffer- 
ence from  the  electrical  standpoint  whether 
the  primary  or  the  secondary  part  rotates.  From  the  mechan- 
ical standpoint  it  is  usually  desirable  to  have  the  secondary  part 
rotate,  as  the  primary  windings,  which  are  usually  of  higher 


Surrounded  by  a Six-pole 
Rotating  Field  of  the 
Primary  Stator. 


826 


ALTERNATING  CURRENTS 


voltage  than  the  secondary,  are  then  subject  to  less  mechanical 
vibration  and  can  be  connected  directly  to  the  supply  mains 
without  the  interposition  of  collector  rings.  Coil-wound  sec- 
ondary windings  are  in  general  of  the  same  type  as  the  arma- 
ture windings  of  synchronous  alternators,  and,  as  explained, 
may  have  their  terminals  short-circuited,  or  they  may  be  con- 
nected (through  collector  rings  if  the  armature  rotates)  to  out- 
side devices  suitable  for  starting  or  for  regulation  purposes.* 
The  primary  windings  of  induction  motors  are  almost  always 
arranged  to  produce  more  than  two  poles,  in  order  to  bring  the 
machine  to  a reasonable  speed.  The  rotating  field  frequency 

/ 

in  revolutions  per  second  is  equal  to  — , where  / is  the  fre- 
quency of  the  alternating  current  and  p the  number  of  pairs  of 
poles  in  the  magnetic  field,  and  in  revolutions  per  minute  this 

PA 

becomes  V=  — — . We  therefore  have  the  following  table  of 

p 6 

motor  speeds  for  the  frequencies  common  in  this  country. 


Relation  of  Number  of  Poles  to  Synchronous  Speed  in  Induction 

Motors 


Number  of  Poles 
of  Motor 

V,  WHEN  / = 25 

Y,  vritEN  f = 60 

2 

1500 

3600 

4 

750 

1800 

6 

500 

1200 

8 

375 

900 

10 

300 

720 

12 

250 

600 

16 

187.5 

450 

20 

150 

360 

24 

125 

300 

This  table  shows  the  futility  of  attempting  to  build  satisfactory 
induction  motors,  intended  for  use  on  even  the  lowest  fre- 
quency commonly  used  in  this  country,  with  less  than  four 
poles  ; and  on  the  higher  frequencies  not  less  than  six  to  ten 
poles  are  required  to  give  reasonable  speeds.  Motors  of 
greater  output  than  ten  horse  power  should  have  a sufficiently 


* Art.  19(5. 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


827 


large  number  of  poles  to  give  a field  velocity  which  does  not 
exceed  750  or  800  revolutions  per  minute.  A 

The  primary  windings,  in  the 
very  early  history  of  the  art, 
were  sometimes  placed  directly 
upon  polar  projections  of  the 
field  frame,  as  in  Fig.  470, 
which  shows  a four-pole  two- 
phase  machine.  In  modern 
machines  the  windings  are  ar- 
ranged as  embedded  uniformly 
distributed  conductors  in  a 
frame  of  uniform  magnetic  sur- 
face, as  in  Fig.  471,  which 
shows  a four-pole  three-phase 
machine.  Conductors  embedded  in  open  slots  give  the  most 
approved  arrangement,  since  embedding  serves  to  reduce  the 


Fig.  471.  — Diagram  of  Rotating  Field  Induction  Motor,  in  Which  the  Stator  has  a 
Tri-phase  Four-pole  Primary  Winding,  and  the  Rotor  may  be  wound  either  as  a 
Squirrel  Cage  or  Coil-wound  Secondary. 


Fig.  470. — Diagram  of  Salient  Pole  In- 
duction Motor  Primary  Winding  as 
sometimes  used  in  the  Early  History  of 
the  Art. 


828 


ALTERNATING  CURRENTS 


reluctance  of  the  magnetic  circuit  and  reduce  magnetic  leakage, 
and  therefore  serves  to  increase  the  power  factor  of  the  motor. 
Open  slots  are  better  than  closed  slots,  as  they  offer  greater 
reluctance  to  the  magnetic  leakage  paths  and  permit  the  use  of 
form-wound  coils.  Since  the  actual  magnet  poles  are  produced 
by  the  resultant  effects  of  the  polyphase  currents,  it  requires  two 
coil  sections  to  produce  each  magnet  pole  in  a two-phase  rna- 


Fig.  472. — Simple  Ring  Diagram  for  showing  the  connections  of  a Four-Pole  Tri- 
phase Wye-Connected  Winding. 

chine,  and  three  coil  sections  in  a three-phase  machine.  The 
connections  of  the  field  coils  may  be  traced  out  according  to  the 
instructions  given  for  connecting  the  armature  coils  of  poly- 
phase generators.* 

The  connections  for  a three-phase  field  winding  are  illustrated 
diagrammatically  by  Fig.  472,  which  shows  a wye-connected, 
four-pole  ring  field.  The  arrows  indicate  how  the  magnetic 
fluxes  combine  at  a particular  instant  under  the  impulsion  of  the 
magneto-motive  forces  of  the  tri-phase  currents.  It  should  be 


* Art.  22. 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


829 


1 1 
• I 

s ! 


NsX> 


Fig.  473. — Diagram  of  a Simple  Progressive  Tri- 
phase  Eight-pole  Winding  such  as  is  to  be  found  on 
the  Primary  and  Secondary  Windings  of  Fig.  474- 


noted  that  the  ring  type  of  winding  is  used  in  this  illustration 
for  the  sake  of  simplicity  in  showing  the  connections.  Wind- 
ings embedded  in  the  polar  surfaces  are  now  used  almost  alto- 
gether, but  the  principle  of  the  connections  is  the  same  as  in  the 
simple  diagram  of  the  figure.  The  wye  connection  is  often 
preferred  for  three-phase  field  windings,  as  less  voltage  is  im- 
pressed on  a coil,  so  that 

fewer  turns  of  wire  are  ' > 
required,  and  the  strain 
on  the  insulation  of  each 
coil  is  less.  The  circu- 
lation of  internally  in- 
duced third  harmonic 
currents  is  also  pre- 
vented. The  total 
weight  of  copper  is 
equal  in  wye  and  delta 
connections.  Figure 
473  shows  a development 

of  the  eight-pole,  wye-connected  winding  such  as  that  of  Fig.  474. 

In  the  case  of  Fig.  474,  the  secondary  winding  terminals 
are  connected  to  rings  so  that  regulating  or  starting  resist- 
ances may  be  inserted. 

It  is  seen  from  the  above  discussion  that  the  primary  wind- 
ings of  rotating  field  induction  motors  are  of  the  same  forms  as 
the  armature  windings  of  synchronous  alternators.  The  second- 
ary windings  may  be  of  the  short-circuited  squirrel  cage  type  or 
be  similar  to  the  primary  windings,  but  the  numbers  of  slots  in 
the  field  and  armature  cores  should  differ,  and  should  be  of  num- 
bers which  are  prime  to  each  other,  to  prevent  recurrent  varia- 
tions of  the  resultant  reluctance  of  the  mutual  magnetic  circuit. 

The  frequency  of  the  magnetic  cycles  in  the  iron  of  the 
primary  core  is  equal  to  the  frequency  of  the  current  flowing 
in  the  magnetizing  coils,  but  in  the  armature  it  is  equal  to 
the  motor  slip ; and  the  hysteresis  and  eddy  current  losses 
per  pound  of  iron  are  therefore  many  times  greater  in  the 
primary  core  than  in  the  secondary  core  when  the  motor  is 
operating  at  ordinary  speeds.  In  this  respect  the  character- 
istics of  an  induction  motor  are  the  reverse  of  those  of  a 
synchronous  alternator,  in  which  the  field  loss  consists  in  most 


830 


ALTERNATING  CURRENTS 


part  of  the  PR  loss,  while  the  armature  loss  is  the  sum  of  the  I2R 
and  core  losses,  of  which  the  latter  may  be  the  larger  portion. 
In  the  induction  motor,  the  field  core  losses  are  large  and  the 


Fig.  474.  — Diagram  of  a Rotary  Field  Induction  Motor  having  Eight  Poles.  The 
Field  and  Armature  Cores  are  both  wound  for  Three  Phases  Wye  connected  and 
the  Secondary  Winding  has  its  Free  Terminals  connected  to  Collector  Rings  for 
Use  with  Starting  and  Regulating  Resistances. 


armature  core  losses  are  scarcely  appreciable  in  a well-designed 
machine;  it  is  therefore  desirable  to  reduce  the  amount  of  iron 
in  the  primary  core  to  a small  volume,  keeping  in  mind  the  effect 
of  the  several  variables  such  as  weight  of  copper,  density  of  flux, 
etc.,  which  enter  in  the  problem  as  they  do  in  the  design  of  a 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


831 


transformer.*  It  will  be  noted  that  with  a rotating  armature 
the  winding  space  is  smaller  for  the  secondary  than  for  the  pri- 
mary windings,  but  the  voltage  of  the  former  is  usually  lower 
than  that  of  the  latter,  so  that  less  space  is  used  for  insulation. 
The  smaller  the  number  of  slots  per  coil,  the  smaller  is  the  re- 
luctance of  the  leakage  paths  and  hence  the  greater  is  the  leak- 
age reactance  with  its  attendant  disadvantages.  It  is  therefore 
common  practice  to  make  the  rotor  diameter  large  and  the  length 
correspondingly  short  so  that  from  3 to  6 slots  can  be  used  on 
the  stator  per  coil  side  and  from  3 to  7 on  the  rotor  — the  num- 
ber depending  upon  the  given  conditions  that  must  be  fulfilled 
in  each  particular  motor  design,  such  as  voltage,  speed,  rated 
load,  etc.  To  reduce  reactance  the  air  space  is  of  course  made 
as  small  as  is  mechanically  practicable. 

In  all  cases,  especially  where  squirrel  cage  secondaries  are 
used,  the  number  of  primary  core  slots  should  be  an  uneven 
multiple  of  the  number  of  secondary  core  slots,  as  intimated, 
in  order  that  there  may  be  no  dead  points  in  starting.  If  the 
teeth  of  the  primary  core  lie  exactly  opposite  the  teeth  of  the 
secondary  core,  the  magnetic  reluctance  of  the  paths  through 
the  teeth  will  be  very  low  compared  with  that  of  the  wind- 
ing spaces  between  the  teeth.  This  causes  exceedingly  dense 
tufts  of  magnetic  flux  to  be  set  up  between  opposing  teeth  at 
the  instant  when  the)''  are  facing  each  other.  The  tendency, 
then,  of  each  tooth  of  an  opposing  pair  is  to  strongly 
attract  the  other  as  the  magnetic  poles  created  by  the  flux 
at  their  ends  are  of  opposite  sign.  If  at  starting  the  tooth 
to  tooth  attraction  is  stronger  than  the  pull  between  the 
secondary  current  and  rotating  magnetic  flux,  the  motor  will 
not  move.  Using  unlike  numbers  of  slots  on  the  two  members 
avoids  this  trouble.  In  a somewhat  similar  manner  when 
the  armature  is  running  unloaded  and  nearly  at  synchronism, 
there  is  a tendency  for  the  secondary  core  to  jump  into  syn- 
chronism with  the  rotating  magnetic  field  and  convert  the 
machine  for  the  time  into  synchronous  motor  relations,  the  ar- 
mature becoming  the  mechanically  rotating  field  magnet  with 
permanent  poles  magnetized  by  the  synchronously  running  re- 
sultant rotating  field.  This  is  because  the  secondary  current 
when  the  speed  nears  synchronism  at  no  load  is  exceedingly 


* Arts.  135-137. 


832 


ALTERNATING  CURRENTS 


small,  only  enough  to  drive  the  armature  against  the  resisting 
moment  of  the  friction  of  bearings  and  air,  and  the  attraction 
between  teeth  under  these  circumstances  sometimes  shows  a 
larger  torque  than  that  between  the  rotating  flux  and  the 
bands  of  armature  current.  When  this  occurs,  the  former 
prevails  and  the  secondary  core  takes  the  same  speed  of  ro- 
tation as  the  rotating  flux.  While  this  condition  continues, 
the  secondary  current  becomes  zero  and  the  windings  act 
somewhat  to  maintain  the  condition  of  synchronism  as  will  be 
seen  later.  The  condition  is  not  necessarily  undesirable,  though 
it  calls  for  a sudden  change  in  primary  current.  Where  a load 
of  any  material  size  comes  on  to  the  motor,  the  secondary  core 
drops  out  of  synchronism  and  the  machine  resumes  its  action 
as  an  induction  motor.  This  peculiarity  is  of  material  value 
when  synchronous  motors  are  started  as  induction  motors.* 

196.  Starting  and  Regulating  Devices.  — Several  different  ar- 
rangements ma}r  be  used  for  starting  polyphase  induction  motors. 

1.  Small  machines  are  commonly  connected  directly  to  the 
circuit  without  the  intervention  of  any  special  starting  devices. 
This  is  not  a safe  proceeding  for  large  machines,  as  when  the 

secondary  circuit  is  at  rest  and  the 
primary  windings  are  directly  con- 
nected to  the  supply  circuit,  the 
machine  is  in  the  condition  of  a 
transformer  with  the  secondary 
windings  short-circuited,  and  is 
liable  to  burn  up  before  getting 
under  way.  Some  of  the  older 
small  motor  armatures  were  ar- 
ranged with  two  rows  of  con- 
ductors, making  two  independent 
squirrel  cages  (Fig.  475),  one  con- 
siderably farther  from  the  armature 
surface  than  the  other  for  the  ostensible  purpose  of  reducing  the 
starting  current  of  the  machines. 

2.  (a)  Resistance  in  Field  Circuit.  — Resistance  may  be  in- 
serted in  the  circuits  leading  to  the  primary  windings  to  be 
used  in  much  the  same  manner  as  starting  rheostats  are  used 
in  starting  direct-current,  constant  voltage  motors.  Rheostats 

* Art.  167. 


Fig.  475.  — Squirrel  Cage  Secondary 
with  Double  Row  of  Conductors. 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


833 


arranged  in  this  way  in  each  circuit  must  be  manipulated  simul- 
taneously in  all  the  circuits,  and  therefore  must  be  mechanically 
coupled.  Besides  starting  rheostats  similar  to  starting  boxes  for 
direct-current  motors,  liquid  rheostats,  arranged  to  be  varied  by 
dipping  plates  in  a bath,  have  been  used  with  early  motors. 
Three  rheostats  are  required  for  a tri-phase  motor,  and  two  for 
a quarter-phase  motor  operated  on  independent  circuits.  Quar- 
ter-phase motors  on  three-wire  circuits  may  be  started  with  a 
single  resistance  device  inserted  in  the  common  wire,  or  by  two 
resistance  devices  inserted  respectively  in  the  independent  wires. 

The  insertion  of  resistance  in  the  field  circuits  of  induction 
motors  serves  to  restrain  the  starting  current  by  reducing  the 
voltage  at  the  primary  terminals.  Tins,  however,  reduces  in 
the  numerator  of  the  expression  * 

T=  K s2j?i2fi2 

(s2/fi  + -Zf2)2  + s22(-^l  + -^2  )2 

and  thereby  greatly  lowers  the  available  torque  per  ampere  and 
the  motor  takes  an  excessive  current  from  the  supply  mains  to 
accelerate  a heavy  load.  In  the  past  this  plan  was  used  quite 
extensively  by  European  manufacturers,  especially  for  large 
machines  which  could  be  started  with  the  belt  on  a loose  pulley. 
This  method  has  the  disadvantage  of  wasting  much  energy 
through  the  dissipation  of  heat  in  the  rheostats,  and  has,  there- 
fore, except  in  special  cases  been  abandoned  in  favor  of  the  auto- 
transformer and  transformer  methods  described  below. 

(6)  Variable  Compensator  or  '•'■Autotransformer.''  — The  vol- 
tage at  the  terminals  of  the  primary  circuits  may  be  reduced  at 
starting  by  introducing  an  impedance  coil  across  the  supply  cir- 
cuits and  feeding  the  motor  from  variable  points  on  its  windings. 
This  arrangement  may  be  caused  to  supply  a large  starting  cur- 
rent to  the  motor  without  interfering  with  the  supply  circuits, 
but  it  has  the  same  effect  on  the  motor  torque  as  resistance  in 
series  with  the  primary  windings.  The  losses  by  this  method  of 
starting  are  much  reduced,  for  the  power  required  by  the  motor 
for  starting  is  delivered  at  the  proper  reduced  voltage  by  trans- 
former action  at  comparatively  high  efficiency.  Figure  476 
shows  a compensator  connected  to  a tri-phase  motor  with  a no- 
voltage release,  which  automatically  opens  the  circuit  when,  for 

* Art.  192. 

3 H 


834 


ALTERNATING  CURRENTS 


any  reason,  the  supply  voltage  disappears.  It  contains  also  an 
overload  release.  The  lower  row  of  terminals,  designated  the 
starting  side,  are  connected  by  proper  switch  blades  or  contactors 


Fig.  476.  — Diagram  of  Compensator  Starting  Device  for  Tri-phase  Induction  Motor, 
with  No-voltage  and  Overload  Release. 

with  the  middle  row  marked  “ cylinder  switch”  when  the  motor 
is  to  be  started.  This  connects  the  motor  with  the  low  voltage 
terminals  of  the  autotransformer  or  compensator  and  the  supply 


Fig.  477.  — Diagram  of  Compensator  Starting  Device  for  a Quarter-phase  Motor  with 
No-voltage  and  Overload  Release. 

mains  with  the  primary  terminals  thereof.  When  the  motor 
has  reached  its  full  speed  for  this  connection,  the  central  switch 
cylinder  blades  or  contactors  are  thrown  over  to  the  running  side 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


835 


and  the  motor  is  connected  directly  to  the  supply  lines.  When 
very  large  motors  are  in  use  several  steps  can  be  made  in  this 
operation  by  having  several  sets  of  taps  on  the  compensator 
coils  and  stepping  from  one  to  the  next  successively.  Figure 
477  shows  a similar  arrangement  for  a quarter-phase  motor. 

It  is  evident  that  these  arrangements  are  purely  starting 
devices  and  not  regulators,  since  from  the  motor  formulas  * it 
is  seen  that  the  running  torque  is  changed  with  the  square  of 
the  voltage;  and,  therefore,  if  it  should  be  desired  to  drive  a 
load  of  given  torque  at  reduced  speed,  reduction  of  the  impressed 
voltage  by  means  of  the  compensator  could  only  change  the 
speed  through  very  narrow  limits  before  the  maximum  motor 
torque  would  be  reduced  to  the  opposing  moment  of  the  load. 
Further  reduction  of  voltage  would  result  in  the  motor  coming 
to  a standstill. 

3.  Resistance  in  Armature  Circuits  for  Speed  Regulation.  — The 
torque  of  the  armature  at  starting  may  be  made  equal  to  the  max- 
imum running  torque  by  inserting  a definite  amount  of  resistance 
in  the  secondary  circuits  which  increases  the  total  armature  re- 


nal or  starting  resistance.  The  value  of  Re  may  be  determined 


where  s2m  is  the  slip  at  maximum  torque,  and  Rt  is  the  total 
resistance  through  which  the  armature  current  flows.  There- 
fore to  get  a maximum  torque  when  s2m  = l,  requires  that 


reactances  are  determined  from  the  circle  diagram  or  by  calcu- 
lation. The  total  resistance  used  in  a starting  rheostat  should 
be  much  larger  and  arrangements  should  be  made  to  reduce  it 
gradually  in  order  that  the  motor  may  not  start  with  a jerk. 

As  the  maximum  torque  is  usually  designed  to  occur  at  a slip 
between  one  and  one  third  and  two  times  that  corresponding  to 
the  normal  full  load  torque,  when  running  with  the  winding 
resistance  only  in  the  secondary  circuits,  the  armature  and  field 
currents  at  maximum  torque  do  not  exceed  twice  the  full  load 
currents,  so  that  resistance  inserted  in  the  armature  circuits 


as  follows:  As  already  shown, f s2m  = 


Rt 


[R*  + (x1  + x2y  ]*, 


R%  + Re 


= 1 or  Re  = [Rf  + (Xx  + X2)2]*  - Rr  The 


[R^  + (X1  + X2nh 


t Art.  192. 


* Art.  192. 


836 


ALTERNATING  CURRENTS 


serves  the  double  purpose  of  increasing  the  starting  torque  and 
keeping  the  starting  current  within  bounds. 

This  plan  is  largely  used  by  many  commercial  companies. 
The  arrangement  of  the  starting  rheostat  depends  largely  upon 
the  type  of  the  secondary  windings  to  which  it  is  applied. 

In  secondary  windings  of  the  squirrel  cage  type  the  conductors 
may  be  tipped  at  one  end  with  tapered  high  resistance  metal 
strips,  which  come  in  contact  with  a sliding  copper  ring.  At 
starting,  this  ring  may  just  touch  the  high  resistance  tips,  and 
as  the  machine  speeds  up  the  ring  may  be  slid  along  until  the 
tips  are  cut  out  and  the  copper  armature  conductors  are  directly 
connected  together  through  the  ring.  If  the  armature  revolves, 
it  is  evident  that  the  ring  must  be  arranged  to  slide  on  a spline 
on  the  shaft,  and  to  be  controlled  by  a grooved  sliding  collar 
and  loose  lever.  The  same  device  may  be  used  for  secondary 
windings  with  coils  having  a common  short-circuiting  point. 
In  this  case  one  set  of  coil  terminals  are  permanently  connected 
together,  and  the  other  set  are  connected  into  a rheostat  of  high 
resistance  metal  strips,  which  may  be  short-circuited  by  a slid- 
ing ring,  as  already  explained. 

If  the  secondary  windings  are  arranged  so  as  to  have  but  one 
coil  for  each  phase,  the  introduction  of  resistance  is  very  simple, 
since  only  one  resistance  coil  for  each  phase  is  required.  In 
this  case,  if  the  secondary  core  is  stationary,  the  connections  of 
the  rheostat  are  made  directly  into  the  secondary  circuits,  or 
between  the  secondary  circuits’  and  one  point  of  common  con- 
nection ; while  if  the  secondary  core  revolves,  collector  rings 
may  be  placed  on  the  shaft,  and  stationary  rheostats  may  be  used 
to  control  the  resistance  of  the  coils  which  are  properly  con- 
nected to  the  rings,  or  the  resistance  may  be  placed  inside  the 
armature  spider  and  be  controlled  by  a sliding  collar  and  loose 
lever.  Figure  478  shows  a tri-phase  motor  having  three  col- 
lector rings  for  connecting  up  external  starting  and  regulating 
resistance  to  the  secondary  windings.  The  rheostat  coils  can 
be  connected  in  wye  or  delta.  A quarter-phase  winding  would 
preferably  use  four  rings  and  two  resistance  devices. 

By  using  this  method  the  motor  can  be  regulated  for  any 
torque  up  to  its  maximum  value  for  all  speeds,  from  standstill 
to  the  speed  at  which  the  machine  drives  its  load  when  the  sec- 
ondary winding  is  short-circuited.  The  method  is  effective  for 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


837 


speed  control,  but  has  the  disadvantage  of  absorbing  much 
power  in  the  external  resistance  when  the  speed  is  made  much 
below  that  normal  for  the  given  load  when  the  secondary  wind- 
ings are  short-circuited  without  external  resistance.  Also, 
with  a given  resistance  inserted,  the  speed  varies  with  the  load. 
These  conditions  are  somewhat  analogous  to  those  of  a direct- 
current  shunt- wound  motor  when  the  speed  is  controlled 
through  the  introduction  of  external  resistance  into  the  arma- 
ture circuit. 


Fig.  478.  — A Rotary  Field  Motor  with  Tri-phase  Secondary  Winding  arranged  with 
Collector  Rings  for  the  Introduction  of  External  Starting  or  Regulating  Re- 
sistances. 

4.  Commutated  Armature. — The  armature  may  be  wound 
with  the  coils  so  arranged  that,  instead  of  starting  with  all 
conductors  in  cumulative  series  in  each  coil,  a portion  of  the 
conductors  are  connected  in  opposition  to  the  others  at  the 
start,  and  are  then  reversed  and  connected  properly  in  series 
with  the  others  after  the  machine  is  in  operation.  The  opposi- 
tion arrangement  affects  the  current  both  of  the  secondary  and 
primary  circuits. 

Again,  the  connectors,  by  means  of  which  the  armature  con- 
ductors are  placed  in  series,  may  be  made  of  high  resistance 
material,  and  these  connectors  may  be  excluded  from  the  cir- 
cuit when  the  conductors  are  short-circuited  together  after  the 


838 


ALTERNATING  CURRENTS 


motor  is  running  at  full  speed.  Devices  of  this  character  can 
be  used  for  giving  two  motor  speeds  through  the  range  of  torque 
for  which  the  motor  is  designed,  but  any  additional  speeds  would 
make  cumbersome  connections. 

5.  Commutated  Primary  Windings.  — In  order  to  enable  a 
motor  to  run  at  two  speeds  without  loss  in  external  secondary 
resistance,  the  number  of  poles  of  the  rotating  magnetic  field 
may  be  changed  by  properly  arranging  the  primary  windings. 
Thus,  a motor  having  a primary  winding  for  eight  poles  may 
have  its  speed  doubled  by  so  connecting  the  primary  coils  that 
the  resultant  flux  has  four  poles.  This,  as  may  be  seen  by 
a study  of  primary  windings,  requires  quite  a little  shifting 
of  coil  terminals,  which  becomes  burdensome  if  the  method 
is  used  for  obtaining  more  than  two  speeds.  To  obtain  the 
same  motor  torque  at  the  double  speed  requires  one  half  as 
many  primary  coils  per  phase,  since  the  number  of  poles  of  the 
rotating  field  is  cut  in  half,  while  the  primary  and  secondary 
magneto-motive  force  must  remain  constant  to  supply  the  neces- 
sary torque,  since  the  flux  per  pole  is  doubled  but  the  polar 
area  is  also  doubled.  Changing  from  wye  to  delta  connection 
for  starting  purposes  is  referred  to  on  page  857. 

6.  Concatenation  Control.  — Where  two  like  motors  are  con- 
nected rigidly  to  the  same  mechanical  load,  as  when  two  poly- 


Fig.  479.  — Diagram  of  Connections  of  Tri-phase  Motors  for  Concatenation  Control. 

phase  motors  are  used  in  driving  an  electric  car,  a speed  control 
may  be  obtained  which  is  somewhat  analogous  to  the  series- 
parallel  control  of  two  direct-current  electric  traction  motors. 
Figure  479  shows  two  motors,  both  having  unity  ratio  of  trans- 
formation between  their  primarjr  and  secondary  windings,  con- 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


839 


nected  in  this  way,  which  is  called  the  Tandem  or  Concatenated 
connection,  ready  to  start.  Motor  A is  connected  directly  to 
the  supply  mains  ; its  secondary  windings  are  connected  to  the 
primary  windings  of  motor  _B,  which  thereby  is  supplied  with 
current  of  frequency  corresponding  to  the  slip  of  motor  A ; the 
secondary  windings  of  B are  connected  to  an  external  set  of 
rheostats  ; and  the  shafts  of  the  two  motors  are  rigidly  joined 
mechanically.  The  ratios  of  transformation  of  the  motors  are 
equal,  and  the  motors  are  otherwise  alike.  When  motor  A 
begins  to  rotate,  motor  B is  caused  to  rotate  through  the 
mechanical  connection.  As  the  resistance  in  the  secondary 
circuits  of  motor  B is  cut  out,  the  speeds  rise  until,  when  the 
resistance  is  short-circuited,  the  motors  are  running  at  some- 
thing under  one  half  the  synchronous  speed  pertaining  to  each 
when  supplied  with  voltage  at  the  frequency  of  the  mains. 
They  cannot  run  above  half  speed  and  carry  any  considerable 
load,  for  in  that  case  motor  B would  run  above  synchronism 
with  the  current  supplied  to  its  primary  windings  from  the  sec- 
ondary windings  of  A , and  this  would  convert  motor  B into  an 
asynchronous  generator  * tending  to  force  .current  back  into  the 
secondary  circuits  of  A.  The  faster  the  motor  armatures  rotate, 
the  smaller  is  the  slip  of  A and  the  lower  the  frequency  of  the 
current  it  supplies  to  B.  At  a speed  of  revolution  equal  to 
one  half  synchronous  speed  for  motor  A , the  rotating  flux  of  B 
and  its  armature  conductors  would  be  in  exact  synchronism. 
Then  the  only  current  that  could  flow  from  the  secondary  cir- 
cuits of  motor  A would  be  that  delivered  to  the  primary  circuits 
of  B , which,  under  the  conditions  named,  would  be  the  exciting 
current  necessary  to  set  up  the  rotating  magnetic  field  of  B. 
Therefore  neither  motor  would  produce  useful  torque  at  this 
speed.  Since  the  resisting  moment  of  the  load  must  be  equaled 
by  the  electrical  torque,  this  state  could  not  exist  when  any 
considerable  load  is  to  be  driven ; and,  therefore,  the  speed  of 
rotation  of  the  armatures  when  driving  a load  must  be  less  than 
one  half  normal  speed  by  an  amount  which  results  in  currents 
flowing  in  the  windings  sufficient  to  produce  the  necessary 
torque.  At  this  point  the  slip  in  B (with  respect  to  the  field 
frequency  provided  by  current  from  the  secondary  circuits  of 
A)  is  sufficient  to  induce  enough  secondary  voltage  to  drive 

* Art.  194. 


840 


ALTERNATING  CURRENTS 


current  through  its  secondary  winding.  The  slip  in  A (with 
respect  to  the  field  frequency  provided  by  current  from  the 
mains)  is  sufficient  to  induce  enough  secondary  voltage  in  its 
secondary  windings  to  drive  the  current  through  the  imped- 
ance of  those  windings  plus  as  much  more  as  is  required  to 
provide  the  primary  terminal  voltage  at  B. 

The  primary  current  in  each  phase  of  A is  the  sum  of  a com- 
ponent equal  and  opposite  to  the  secondary  current  of  B,  the 
exciting  current  of  B , and  the  exciting  current  of  A.  The 
currents  in  the  secondary  circuits  of  A are  equal  to  those  in 
the  primary  circuits  of  B.  The  primary  voltage  is  absorbed  by 
the  drop  due  to  the  mutually  induced  voltage  of  A,  the  imped- 
ance of  the  primary  and  secondary  windings  of  A,  the  mutually 
induced  voltage  in  B , and  the  impedance  of  the  primary  and 
secondary  windings  of  B. 

The  induction  motor  locus  diagram  given  in  Fig.  463  can  be 
readily  applied  to  this  case  by  letting  the  current  locus  which 
represents  the  current  in  the  secondary  windings  of  B have  a 
diameter  equal  to  the  voltage  of  the  mains  divided  by  the  sum 
of  the  leakage  reactances  of  the  two  motors,  and  the  exciting 
current  have  a value  equal  to  the  combination  of  the  exciting 
currents  of  the  two  motors.  The  quadrature  voltage  in  the 
voltage  locus  diagram  of  Fig.  463  is  then  the  sum  of  the 
leakage  drop  voltages ; and  the  active  voltage  is  the  sum  of 
the  drops  through  the  counter-voltages  and  resistances  of  the 
two  motors.  If  the  ratios  of  transformation  of  the  motors  are 
not  both  exactly  alike  or  the  impedances  of  the  two  motors  are 
not  equal,  one  motor  will  “ rob  ” the  other  of  load  with  danger 
of  its  becoming  overloaded;  which  is  comparable  to  the  results 
in  the  case  of  two  rigidly  connected  series  direct-current  motors 
which  have  different  armature  resistances  or  field  strengths. 

As  a result  of  this  discussion  the  conclusion  can  be  safety 
drawn  that  the  rotary  field  induction  motor  does  not,  in  its 
present  form,  lend  itself  readily  to  wide,  economical  variation 
of  speed.  Where  such  speed  variation  is  demanded,  series  or 
repulsion  motors,  which  are  described  in  later  pages,  are  more 
naturally  fitted. 

197.  Reversing  Polyphase  Motors.  — Potyphase  motors  may 
be  reversed  by  reversing  the  direction  of  rotation  of  the  rotat- 
ing magnetic  field. 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


841 


In  quarter-phase  motors  with  two  independent  circuits,  re- 
versing the  terminal  connections  of  either  circuit  will  effect  the 


reversal  of  rotation,  but  reversing  the  terminals  of  both  circuits 
will  not  alter  the  direction  of  rotation.  Quarter-phase  motors 


Fig.  480.  — Curves  for  showing  the  Effect  of  Third  and  Fifth  Harmonics  on  the  Operation  of  a Quarter-phase  Induction  Motor. 


842 


ALTERNATING  CURRENTS 


with  three-wire  connections  may  be  reversed  by  interchanging 
the  connections  of  the  outside  conductors,  leaving  the  common 
return  conductor  unchanged. 

The  direction  of  rotation  of  tri-phase  motors  may  be  reversed 
by  interchanging  the  connections  of  any  pair  of  leads. 

198.  Effect  of  the  Form  of  Curves  of  Voltage.  — The  effect  of 
distorted  curves  of  voltage  upon  the  operation  of  induction 
motors  depends  upon  the  number  of  phases.  The  harmonics  of 
three  and  five  times  the  fundamental  frequency  are  the  only 
ones  which  need  be  considered ; and  indeed,  that  of  three  times  * 
the  fundamental  frequency  is  the  only  one  which  as  a rule  has 
an  appreciable  influence.  In  single-phase  motors  the  harmonics 
should  affect  the  magnetic  field  as  they  affect  that  of  a trans- 
former, so  that  peaked  voltage  curves  should  cause  a decrease  in 
core  losses,  and  the  operation  of  the  motor  should  not  be  other- 
wise greatly  influenced.  In  polyphase  motors,  however,  the 
harmonics  may  set  up  a rotating  field  of  their  own,  which  is 
superposed  upon  the  regular  held,  and  may  interfere  with  the 
operation  of  the  machine. 

The  harmonics  with  three  times  the  fundamental  frequency 
belonging  to  the  two  circuits  of  a quarter-phase  system  have  a 
phase  difference  of  90°  (Fig.  480),  and  these  set  up  a superposed 
rotating  field  in  the  induction  motor  which  has  a field  velocity 
of  three  times  that  of  the  main  field.  The  figure  shows  that 
the  harmonics  of  ti'iple  frequency,  belonging  to  the  two  phases, 
are  reversed  in  relative  position  compared  with  the  fundamental 
waves.  The  field  due  to  these  harmonics  rotates  in  the  reverse 
direction  from  that  of  the  main  field,  and  therefore  tends 
directly  to  decrease  the  torque  of  the  motor  and  to  increase  the 
slip.  The  field  due  to  the  harmonics  of  five  times  the  frequency 
rotates  in  the  same  direction  as  the  main  field,  and  its  only  dis- 
advantageous effect  is  in  causing  eddy  currents  which  may 
slightly  decrease  the  efficiency  of  the  motor.  The  frequencies 
of  the  harmonic  curves  are  indicated  in  the  figure  by  subscripts. 

In  tri-phase  circuits  the  harmonics  of  triple  frequency  belong- 
ing to  the  different  currents  are  directly  superposed  in  phase  as 
is  shown  in  Fig.  481,  and  therefore  the  superposed  field  which 
they  cause  in  tri-phase  induction  motors  is  a stationary  one 
whose  influence  is  only  to  decrease  the  efficiency  by  setting  up 
extra  losses.  The  figure  also  shows  that  the  harmonics  of  five 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


843 


frequencies  have  120°  difference  of  phase  and  are  in  reversed 
orders,  so  that  they  set  up  a reverse  rotating  field,  and  if  they 
are  of  much  strength  they  may  affect  the  torque. 


199.  Single-phase  Induction  Motors. — If  the  field  magnet  of 
an  induction  motor  is  wound  with  one  set  of  coils  so  that  the 


Fig.  481.  — Curve  showing  the  Effect  of  Third  and  Fifth  Harmonics  on  the  Operation  of  a Tri-phase  Induction  Motor. 


844 


ALTERNATING  CURRENTS 


field  poles  are  set  up  by  a single  alternating  current  flowing  in 
the  coils,  the  poles  will  be  stationary  but  alternating,  and  the 
effects  of  electro-magnetic  repulsion  may  be  utilized  for  the  pur- 
pose of  causing  the  armature  to  rotate.  Thus,  if  a uniformly 
wound  short-circuited  armature  (such  as  is  used  for  polyphase 
induction  motors)  is  started  to  revolving  in  a single-phase 
alternating  magnetic  field,  the  balance  of  repulsions  which  ex- 
ists when  the  armature  is  at  rest  is  disturbed,  and  the  armature 
tends  to  continue  its  motion.  To  illustrate  this,  the  condition, 
of  two  coils  in  complementary  positions  with  reference  to  one  of 
the  poles  may  be  considered.  As  the  armature  revolves,  one 
coil  moves  toward  a position  where  it  includes  more  lines  of  force 
from  the  pole,  and  the  other  coil  moves  so  as  to  exclude  lines  of 
force.  If  the  strength  of  pole  is  rising,  the  first  coil  will  have 
the  larger  current  induced  in  it,  since  the  rate  of  change  of  lines 
of  force  through  the  first  coil  is  proportional  to  the  sum  of  the 
rate  of  change  in  the  strength  of  field  and  the  rate  of  change  of 
flux  linkages  due  to  the  motion,  while  the  rate  of  change  of  lines 
of  force  through  the  second  coil  is  the  difference  of  these  two 
rates.  Thanks  to  the  lag  in  the  coil  circuits,  the  currents  in 
both  coils  are  in  such  a direction  as  to  result  in  an  attractive 
force  on  the  pole,  but  a much  stronger  force  is  experienced  by  the 
first  coil  than  the  second.  When  the  field  is  falling,  the  mag- 
netic condition  of  the  second  coil  is  changing  the  more  rapidly, 
but  the  direction  of  the  induced  cui’rents  in  the  coils  is  reversed 
with  respect  to  the  direction  of  the  inducing  field,  and  the  coils 
experience  a repulsive  force  with  reference  to  the  pole.  The 
effect  during  one  complete  period  of  the  magnetic  field  therefore 
tends  to  cause  the  armature  to  rotate  in  the  same  direction  in 
which  it  was  started.  The  torque  is  a maximum  when  the  posi- 
tive product  of  current  and  magnetic  flux  is  a maximum,  which 
is  when  the  current  lags  behind  the  induced  voltage  by  an  angle 
between  45°  and  90°.  The  torque  at  an)'  speed  is  equal  to 
the  torque  which  would  be  given  by  a polyphase  induction 
motor  of  similar  construction  at  the  slip  v = V — V minus  the 
torque  which  the  polyphase  induction  motor  would  give  in  a 
field  of  double  the  frequency,  and  with  a slip  proportioned  to 
V + Tr'=2Tr—v;  where  Y is  synchronous  speed,  V'  actual 
speed,  and  v relative  speed  in  revolutions  per  minute  ; but  with 
the  ordinary  ratio  of  the  resistance  and  inductance  in  the  arma- 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


845 


ture  winding,  the  torque  due  to  the  latter  slip  is  negligible  at 
such  full  load  slips  as  are  satisfactory  in  practice.  In  this  case, 
V need  not  be  looked  upon  as  a speed  of  rotation  of  a magnetic 
field,  but  as  the  speed  of  the  armature  which  keeps  each  con- 
ductor at  the  same  position  with  reference  to  a pole  for  any  fixed 

instant  in  each  period  of  the  magnetic  flux.  Hence  V — 

P 

exactly  as  in  rotary  field  machines.  When  the  armature  is 
t stationary,  v = V,  and  the  two  torques  are  equal  and  opposite. 
A single-phase  induction  motor  may  therefore  be  designed  in  a 
manner  similar  to  the  manner  of  designing  a polyphaser  in  re- 
spect to  its  operation  after  it  has  reached  its  normal  speed,  but 
it  requires  special  treatment  in  the  design  for  the  purpose  of 
making  it  self-starting,  and  it  is  both  heavier  and  less  efficient 
for  a given  output. 

The  principal  voltages  induced  are  indicated  in  the  dia- 
gram of  Fig.  482  for  a bi-polar  machine.  The  alternating 
magneto-motive  force 
impressed  by  current 
flowing  from  the  ex- 
ternal circuit  through 
the  primary  or  field 
coils  is  represented  by 
the  arrow  CD  and  the 
designation  N-S  and 
S-JV.  The  resulting 
alternating  magnetic 
flux  set  up  through 
the  armature  core  sets 
up  counter-voltages  in 
the  secondary  or  arma- 
ture windings,  as  in 
any  transformer.  The 
direction  and  position 
of  the  voltage  in  the  armature  conductors  are  indicated  by  the 
arrow  points  (dots)  and  feathers  (crosses)  marked  in  the  figure 
within  the  cross-sections  of  the  conductors  which  are  indicated 
by  small  circles.  This  voltage  is  in  time  quadrature  with  the 
primary  flux.  These  conditions  are  as  in  a transformer  with 
large  magnetic  leakage  and  may  be  indicated  by  the  ordinary 


Fig.  482.  — Diagram  for  indicating  the  Voltage  gen- 
erated in  a Single-phase  Induction  Motor. 


846 


ALTERNATING  CURRENTS 


transformer  diagram.*  The  turns  across  the  vertical  diam- 
eter have  the  highest  voltage  induced  in  them.  (The  arma- 
ture, for  convenience,  is  assumed  to  be  of  the  squirrel  cage 
type  and  the  primary  windings  to  be  distributed  windings, 
passing  through  the  holes  from  the  top  to  the  bottom  of  the 
figure.)  This  voltage  decreases  more  or  less  sinusoidally  until 
the  horizontal  turns  are  reached,  which  are  in  a neutral  posi- 
tion with  regard  to  N-S.  When  the  armature  revolves,  vol- 
tages as  shown  by  the  arrowheads  and  feathers  indicated  in 
the  inner  circle  are  set  up  in  the  conductors  by  their  mo- 
tion through  the  primary  field.  These  voltages  are  propor- 
tional in  strength  to  the  field  N-S  and  hence  are  in  time  phase 
therewith.  They  reach  a maximum  value  in  the  conductors 
under  the  poles  N-S  and  S-N  and  decrease  more  or  less  sinus- 
oidally to  zero  in  the  turns  90°  therefrom,  which  do  not  cut 
the  field  flux.  There  are,  therefore,  two  sets  of  voltages  in  the 
conductors,  the  first  created  by  ordinary  transformer  action 
caused  by  variation  of  the  field  N-S,  and  the  other  by  the 
motion  of  the  conductors  through  the  lines  of  force  of  the  same 
field.  These  two  voltages  are  at  ninety  degrees  displacement 
with  reference  to  each  other,  both  in  time  phase  and  space 
position.  Two  components  of  current,  conjointly  composing 
the  armature  current,  will  flow  under  the  influence  of  these 
two  voltages.  The  fii’st  reacts  on  the  primary  circuit  like  the 
secondary  current  of  a transformer,  and  has  a time  lag  due  to 
leakage  flux  only  ; while  the  second  or  “ speed  ” current  sets 
up  magnet  poles  at  n-s  and  s-n , and  as  the  reluctance  of  the 
magnetic  path  is  low,  this  component  tends  to  take  a time  quad- 
rature relation  with  its  inducing’  voltage.  Hence  JY-S  and  n-s 
tend  to  rise  and  fall  nearly  in  time  quadrature  and  exactly  in 
space  quadrature. 

When  the  armature  is  running  at  the  speed  corresponding  to 
theoretical  synchronism  for  the  frequency  of  the  primary  cur- 
rent, each  turn  of  wire  on  the  armature  in  each  revolution 
cuts  the  same  number  of  lines  of  force  as  are  comprised  in  the 
maximum  value  of  the  field  JY-S  in  one  cycle  of  the  magnetic 
flux,  so  that  the  transformer  and  speed  voltages  are  equal. 
Therefore,  the  fields  JY-S  and  n-s  are  then  of  approximately 
equal  magnitude.  Under  these  conditions  N-S  and  n-s  may  be 

* Arts.  124  and  129. 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


847 


considered  to  form  an  ordinary  rotating  field  acting  upon  a short- 
circuited  armature,  as  in  polyphase  motors.  When  the  speed 
is  lower  than  the  theoretical  synchronous  speed,  the  speed  vol- 
tage is  proportionately  less;  and  at  standstill  the  speed  voltage 
is  zero. 

The  foregoing  shows  that  it  is  possible  to  consider  the  single- 
phase motor  in  the  light  of  a motor-generator  in  which  the 
transformer  component  of  current  is  a torque  producing  current 
acted  on  by  the  poles  n-s , and  the  speed  current  is  a generated 
exciting  current  which  sets  up  the  motor  fields  n-s.  The 
counter-voltage  caused  by  the  conductors  cutting  flux  n-s  tends 
to  oppose  the  effective  transformer  voltage  in  driving  current 
through  the  armature  resistance.  The  product  of  counter- 
voltage with  transformer  current  equals  the  output  of  the 
motor.  If  the  load  is  changed,  the  motor  speed  must  change 
so  that  the  difference  between  the  transformer  and  counter 
voltages  will  permit  a transformer  current  to  flow  which  will 
give  the  requisite  torque. 

Again,  the  single-phase  alternating  field  may  be  treated  in  a 
different  manner  to  get  the  same  result  more  simply.  An 
alternating  field  stationary  in  position  may  be  resolved  into  two 
rotary  fields,  revolving  in  opposite  directions,  having  the  same 
frequency  as  the  stationary  field,  and  of  one  half  its  magnitude 
in  strength.*  This  is  similar  to  the  principle  of  mechanics  by 
which  a simple  harmonic  motion  may  be  resolved  into  two  uni- 
form opposite  circular  motions  of  one  half  the  amplitude.  The 
torque  diagrams  of  each  of  these  fields  acting  alone  are  shown 
in  Fig.  483,  where  0 is  the  point  of  armature  rest,  and  arma- 
ture speed  is  counted  from  that  point  along  the  horizontal  axis. 
The  curves  A and  A'  are  the  torque  curves  that  would  be  given 
by  either  field  acting  alone,  the  torque  due  to  one  being  in  one 
direction,  and  that  of  the  other  in  the  opposite  direction.  It  is 
evident  that  when  the  armature  is  at  rest  it  has  no  tendency  to 
revolve,  as  the  slip  of  the  armature  with  respect  to  the  two 
fields  is  equal,  and  torques  Ot  and  Ot'  created  by  the  two  fields 
are  equal  and  opposite  ; but  if  the  armature  is  started  in  one 
direction,  for  instance  toward  the  right,  the  slip  with  respect  to 
field  A decreases,  the  torque  caused  by  it  increases  and  tends 

* Ferraris,  A Method  for  the  Treatment  of  Rotating  or  Alternating  Vectors, 
London  Electrician,  Vol.  33,  p.  110. 


848 


ALTERNATING  CURRENTS 


to  continue  the  rotation,  while  the  slip  with  respect  to  field  A' 
increases,  and  the  torque  caused  by  A!  decreases.  When  the 
armature  speed  becomes  V,  the  torque  caused  by  A is  T,  which 
is  due  to  a slip  V—  V=v;  while  the  torque  caused  by  A'  is 
T\  which  is  due  to  a slip  in  relative  speed  between  armature 
and  field  of  V+  V = 2 V—  v.  From  the  relations  of  torque  to 
slip,  which  have  already  been  discussed,*  it  is  evident  that  the 
torque  caused  by  A'  decreases  as  the  relative  speed  increases. 
If  the  differences  between  the  corresponding  ordinates  of  the 
curves  of  torque  A and  A!  are  plotted  in  a curve,  the  actual 


Fig.  483.  — Motor  Torque  Diagram  of  a Single-phase  Motor  with  the  Field  resolved 
into  Two  Component  Rotating  Fields  having  Opposite  Directions  of  Rotation. 

torque  curve  M is  given.  The  ordinates  of  this  will  give  the 
actual  motor  torque  with  respect  to  slip.  From  this  curve  it 
is  seen  that  the  motor  will  work  at  no  load  at  an  almost  syn- 
chronous speed,  and  may  then  be  loaded  until  the  speed  has 
dropped  to  a point  where  the  torque  is  at  a maximum.  If  the 
load  exceeds  this,  the  motor  will  stop.  The  curve  M falls  to  zero 
a little  to  the  left  of  4-  T(  because  of  the  torque  T' . However, 
as  the  backward  field  is,  at  synchronism,  cutting  the  armature 
conductors  at  a rate  of  S'2  = 2 /,  the  reactance  (2  fX2)  against 
that  field  is  very  great  and  the  ordinates  of  T'  (exaggerated  in 
the  figure)  are  ordinarily  negligible  for  working  speeds.  From 

* Art.  192. 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


849 


the  point  + F to  + V there  should  be  a double  frequency  com- 
ponent of  current  in  the  armature  and  a slight  negative  torque. 
The  ordinary  values  of  resistance  and  inductance  which  are  re- 
quired in  an  efficient  and  economical  design,  make  the  effect  of 
A'  so  small  at  the  speed  of  normal  full  load  that  the  action 
of  A only  need  be  considered. 

If  the  armature  should  be  started  toward  the  left,  instead  of 
the  right,  as  here  assumed,  the  conditions  would  be  reversed 
and  the  motor  would  operate  under  the  torque  line  M'.  Evi- 
dently the  formula  from  which  the  torque  curve  M (Fig.  483) 
can  be  plotted  is 

T — K •b^’r^-2 1_ 

C^l  + ^Y)  2 + s22  (^1  + -Y2)2 

_ X {-f  — h)  Ffp/Y 

(2/Bj  ~h^i  + ^2)2  + (2/-s2)2(X1  + X2)2 

The  second  term  becomes  small  for  armature  speeds  approaching 
synchronism.  It  will  be  appreciated,  on  account  of  the  high 
reactance  (2  /—  s2)X2,  that  the  magnetic  flux  of  this  field  which 
actually  passes  through  the  armature  windings  is  very  small 
when  s2  is  small,  so  that  if  <£>'  represents  this  flux,  A>'  = <3?  — <Eq', 
when  <l>  is  the  backward  rotating  component  of  the  flux  in  the 
field,  and  <Eq'  is  the  leakage  component  of  that  flux.  Evidently 
<E>£'  is  always  large  and  approaches  closely  to  the  value  of  <1> 
when  s2  is  small. 

From  these  considerations  it  is  seen  that  a single-phase  motor 
may  be  designed  in  the  same  manner  as  a polyphaser,  and  that 
for  equal  output  the  ampere-turns  upon  the  field  magnet  must 
be  equal  to  the  resultant  number  on  a polyphase  motor.  The 
field  winding  may  be  arranged  as  in  polyphase  motors,  cover- 
ing the  entire  polar  surface,  as  is  shown  in  Fig.  484,  for  a four- 
pole  machine;  but  the  differential  action  in  this  case  reduces 

2 

the  effectiveness  of  the  winding  in  the  proportion  of  1 : as 

7 r 

lias  already  been  shown  in  Art.  188.  Consequently,  the  equa- 
tion from  which  the  field  windings  are  determined  becomes 

xr/  _ x-V2  7 rn'A>f_  2 V2  vn'^f  _ 2 V2 

— 108  108 

The  value  of  K may  be  increased  and  material  saved  by  leaving 
3 1 


850 


ALTERNATING  CURRENTS 


space  between  the  primary  coils,  as  in  Fig.  485,  which  shows 
the  windings  for  a two-pole  field. 

The  efficiency  of  single-phasers  is  less  than  that  of  poly, 
phasers,  since  the  armature  core  losses  are  proportional  to  the 


llllllllllllll 

M 

. 

T 

T i 

Fig.  484. — Diagram  of  a Single-phase  Induction  Motor  with  Primary  Winding  dis- 
tributed over  the  Entire  Surface  of  the  Field  Core. 

frequency  of  the  main  field  instead  of  to  the  slip,  and  the  I2R 
losses  are  greater  ; their  slip  for  a given  load  and  similar  de- 
sign is  greater;  and  their  maximum  torque  is  slightly  less 
than  that  of  polyphasers,  as  is  shown  by  Fig.  483  ; but  these 


Fig.  485. — Diagram  of  a Single-phase  Induction  Motor  with  the  Primary  Winding 
Grouped  in  Coils  covering  Part  of  the  Field  Core. 

differences  in  well-designed  machines  should  not  be  great. 
The  weight  of  single-phasers  is  larger  than  that  of  equal  poly- 
phasers, because  the  value  of  K is  smaller. 

200.  Locus  Diagram  of  the  Single-phase  Motor  and  Substi- 
tuted Impedance. — As  the  single-phase  motor  is  equivalent  to 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


851 


a transformer  having  constant  reactance  and  varying  load  of 
unity  power  factor,  it  may  be  represented  by  a current  locus 
diagram  similar  to  that  of  a transformer  or  polyphase  induction 
motor.  Thus  Fig.  486  shows  such  a diagram,  constructed  and 
lettered  similarly  to  the  diagram  of  Fig.  463,  which  represents 
the  condition  of  one  branch  or  phase  in  a polyphase  induction 
motor.  The  only  important  element  of  difference  is  found  in 


Fig.  486.  — Locus  Diagram  of  a Single-phase  Induction  Motor. 


the  no  load  current  SO.  This  shows  an  active  component 
which  is  greater  in  proportion,  due  to  the  larger  armature 
iron  losses,  and  a wattless  current  which  is  greater  in  propor- 
tion, due  to  the  backward  rotating  components  of  the  field. 

The  line  OF'  is  as  before  the  added  component  of  secondary 
current  due  to  the  load,  but  the  total  secondary  current  is 
greater  than  this  by  some  component  SS'  because  of  the  arma- 
ture current  set  up  by  the  backward  rotating  component  of  the 
magnetic  flux.  Referring  to  Fig.  483,  the  total  armature  cur- 
rent found  by  adding  the  forward  and  backward  components  of 
the  current  may  be  roughly  considered,  for  a given  load,  to  have 


852 


ALTERNATING  CURRENTS 


the  value  F'S'  (Fig.  486),  and  to  have  the  relations  to  the  im- 
pressed voltage  OF,  the  total  primary  current  SF',  and  the  phase 
angles  which  are  shown  in  the  diagram.  This  is  under  the 
assumption  that  the  backward  rotating  component  of  the  primary 
current  remains  constantly  of  uniform  value,  since  its  function  is 
to  set  up  the  backward  rotating  component  of  the  field  magnetic 
flux  <!>'.  The  forward  rotating  component  of  the  field  current, 
on  the  other  hand,  must  not  only  have  a no-load  component  S'  0 
sufficient  to  set  up  its  component  of  the  rotating  flux  <E>,  but 
must  also  neutralize  the  mutual  magnetic  effect  of  the  armature 
current  component  OF',  which  is  the  t.orque-producing  com- 
ponent of  the  armature.  Upon  the  same  suppositions  as  used  in 
the  diagram  of  Fig.  466,  where  the  armature  current  and  voltage 
were  considered  to  be  in  phase  with  each  other,  F' H is  propor- 
tional to  the  flux  necessary  to  induce  the  voltage  which  drives 
the  current  F'O  through  the  armature.  The  diameter  of  the 
locus  semicircle  is  found  in  the  same  way  as  for  the  semicircle 
representing  a phase  of  a polyphase  motor,  as  are  the  various 
elements  of  performance.  KJ  represents  the  slip  line  con- 
structed as  in  Fig.  466;  the  loci  of  copper  loss  may  also  be 
shown  as  in  that  figure. 

Impedances  can  be  substituted  for  the  single-phase  motor 
circuits,  for  the  purpose  of  making  computations,  as  was  done 
with  the  polyphase  motor.*  Referring  to  Fig.  464,  the  shunt 
circuit  representing  the  no-load  losses  should  be  connected  at 
cd  instead  of  ab , in  order  that  great  accuracy  may  be  obtained 
in  representing  the  conditions  of  a polyphase  motor.  In  the 
single-phase  motor  a similar  arrangement  is  required  for  great 
accuracy,  and  allowance  must  be  made  for  the  extra  reactive 
effects  of  the  exciting  currents  ; but  for  most  purposes  it  is  suf- 
ficiently accurate  to  have  the  no-load  circuit  connected  from  a 
to  b , which  will  take  a current  equal  in  phase  and  quantity  to 
the  no-load  current,  and  it  is  upon  the  basis  of  such  an  approx- 
imation that  the  locus  diagram  Fig.  486  is  drawn. 

201.  Starting  Single-phase  Induction  Motors.  — Since  single- 
phase induction  motors  are  not  per  se  self-starting,  special  start- 
ing devices  must  be  included  in  their  design  and  construction. 
This  sometimes  takes  the  form  of  what  is  called  a Phase  splitter. 
The  field  is  wound  with  two  coils  similar  to  the  windings  of  a 

* Art.  192. 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


853 


two-phaser,  and  at  starting  these  are  connected  in  parallel  to 
the  circuit,  one  directly,  and  the  other  through  dead  resistance 
or  capacity.  This  throws  the  currents  in  the  two  coils  into  a 
difference  of  phase,  which  may  be  accentuated  by  winding  one 
coil  so  that  it  has  greater  self-inductance  than  the  other  ; and 
the  machine  then  starts  as  a two-phaser.  After  the  machine  is 
running,  both  coils  are  connected  directly  to  the  circuit,  or  one 
coil  is  cut  out,  and  the  motor  operates  as  a single-phaser. 
This  operation  of  “ phase  spliting,”  as  applicable  to  such  motors, 
cannot  give  a large  difference  of  phase  between  the  currents  in 
the  two  motor  circuits  with  a reasonably  large  power  factor,  and 
consequently  single-phase  induction  motors  started  in  this  way 
must  have  either  a very  small  starting  torque  or  an  unreasonably 
small  power  factor  at  starting.  Another  method  frequently 
used  in  small  motors,  such  as  small  fan  motors,  is  to  use  Shading 
coils.  That  is,  the  motor  field  magnet  may  be  constructed  with 
salient  pole  pieces  like  an  alternator  field  magnet,  — the  core  of 
course  being  laminated.  Around  a similar  tip  of  each  pole  piece 
is  wound  a short-circuited  coil.  The  currents  in  the  short- 
circuited  coils  tend  to  oppose  the  growth  of  magnetic  flux  under 
these  pole  tips,  and  hence  it  rises  to  a maximum  first  in  the  un- 
wound tips,  and  then  in  the  wound  tips,  causing  the  effect  of  a 
two-phase  winding. 

A more  satisfactory  method  is  to  start  the  motor  as  a repul- 
sion motor,*  but  this  requires  a commutator  and  more  expensive 
mechanism.  When  this  plan  is  adopted,  the  armature  is  wound 
with  an  ordinary  reentrant  direct-current  winding  connected 
in  the  usual  manner  to  a commutator,  and  it  is  self-starting 
with  large  torque  when  a short-circuiting  connection  joins  the 
brushes  and  the  brushes  are  displaced  from  the  neutral  plane. 
When  the  armature  has  come  to  full  speed,  an  automatic  device 
may  be  used  to  short-circuit  the  commutator,  and  the  machine 
will  then  run  as  an  induction  motor. 

202.  Efficiency  of  Induction  Motors  and  Methods  of  making 
Tests.  — Polyphase  induction  motors  can  be  built  to  give  about 
the  same  efficiency  as  direct-current  motors,  and  for  somewhat 
less  cost  on  account  of  the  absence  of  a commutator  and  the  low 
insulation  required  on  the  armature  conductors,  but  with  a 
counterbalancing  extra  cost  on  account  of  the  high  grade  and 

* Art.  205. 


854 


ALTERNATING  CURRENTS 


expensive  sheet-iron  stampings  which  are  required  for  the  field 
magnet.  The  most  satisfactory  design,  as  already  explained, 

calls  for  a machine  of 
short  axial  length  and 
large  diameter  in  order 
that  sufficient  teeth 
may  be  used  to  reduce 
magnetic  leakage  to 
small  proportions. 
This  is  well  illustrated 
in  Fig.  487,  which  is 
from  the  photograph  of 
the  armature  of  a 300 
horse-power,  25-cycle, 
10-pole  motor  of  stand- 
ard design,  having  a 
speed  of  300  revolu- 
tions per  minute.  The 
length  of  the  core  is  quite  small  compared  to  its  diameter. 

1.  By  far  the  quickest  method  of  experimentally  obtaining 
the  operating  characteristics  of  an  induction  motor,  including 
efficiency,  torque,  regulation,  and  power  factor,  is  by  means  of 
the  circle  or  locus  diagram.*  The  no-load  exciting  current 
can  be  obtained  in  phase  position  and  scalar  value  by  means  of 
amperemeters,  voltmeters,  and  wattmeters  when  the  armature 
is  running  free.  The  current  locus  is  fixed  with  approximate 
accuracy  by  using  the  same  instruments  to  determine  the  phase 
position  and  quantity  of  the  standstill  current,  using  a reduced 
voltage  for  the  measurements  and  considering  the  current  to  be 
proportional  to  voltage  (or  by  obtaining  it  more  accurately  from 
the  reactances  at  full  load).  The  vector  difference  between  the 
standstill  and  the  no-load  current  is  approximately  the  arma- 
ture standstill  current.  The  length  and  position  of  the  vector 
of  the  armature  standstill  current  determines  the  current  locus. 
The  resistance  of  the  primary  circuit  may  be  measured,  and 
that  of  the  secondary  circuit  may  he  obtained  by  dividing  the 
primary  induced  voltage  (reduced  to  the  secondary  circuit  by 
the  ratio  of  transformation)  by  the  product  of  the  computed 
armature  standstill  current  and  the  cosine  of  its  angle  of  lag. 

* Arts.  191,  192. 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


855 


Correction  can  be  made  for  variations  in  the  exciting  current  if 
great  accuracy  is  desired.  Having  thus  laid  out  the  locus,  the 
desired  operating  quantities  may  be  read  off  for  any  load. 

2.  Direct  Measurement.  In  testing  the  efficiency  of  these 
motors,  the  output  may  be  measured  by  a brake  or  transmis- 
sion dynamometer,  but  the  input  must  be  measured  by  one  of 
the  wattmeter  methods  explained  earlier.  The  two-wattmeter 
method  is  the  best,  but  care  must  be  taken  to  determine  whether 
the  readings  of  the  two  instruments  are  additive  or  subtractive, 
since  the  power  factor  of  a partially  loaded  induction  motor  is 
likely  to  be  quite  low  and  at  no  load  may  be  only  a few  per 
cent.  The  power  factor  is  determined  by  taking  simultaneous 
readings  of  amperemeter,  voltmeter,  and  wattmeter  in  one  cir- 
cuit, if  the  machine  is  balanced ; but  if  the  circuits  differ,  read- 
ings for  each  circuit  must  be  taken.  The  power  factor  is  then 
the  true  watts  divided  by  the  apparent  watts.  This  method 
requires  that  the  motor  shall  be  operated  with  its  full  load  if 
the  full  load  efficiency  is  to  be  obtained,  and  therefore  may 
prove  inconvenient,  and  it  does  not  give  any  way  of  separating 
the  losses. 

3.  Stray  Power  Method.  — A method  similar  to  that  described 
for  testing  transformers  (Art.  147)  is  often  more  convenient  and 
satisfactory.  By  this  plan  the  core  losses,  friction,  and  wind- 
age losses  are  determined  bjr  measuring  by  wattmeter  the  power 
which  is  absorbed  by  the-  motor  when  running  light  under  nor- 
mal voltage  and  frequency.  The  losses  may  be  of  such  value 
that  the  field  current  flowing  is  considerable,  especially  as  the 
power  factor  is  likely  to  be  rather  low,  and  the  PR  loss  cannot 
be  neglected,  but  a correction  can  be  made  after  the  test  for 
copper  losses  is  completed.  To  measure  the  copper  losses,  the 
machine  is  locked  so  as  to  remain  stationary,  in  which  case  the 
armature  serves  the  purpose  of  a short-circuited  secondary  cir- 
cuit, and  such  a reduced  impressed  voltage  is  applied  as  to  cause 
any  desired  current  to  flow  in  the  field  winding.  The  wattmeter 
readings  give  the  PR  losses  for  the  current  flowing,  and  the 
losses  for  any  other  current  may  be  at  once  calculated.  A cer- 
tain amount  of  core  loss  is  included  in  this  measurement,  but 
an  approximate  correction  may  be  made  on  account  of  it  by 
considering  its  ratio  to  the  total  corrected  core  losses  as  the 
1.6  power  of  the  voltage  applied  in  the  copper  loss  test  is  to  the 


856 


ALTERNATING  CURRENTS 


1.6  power  of  the  normal  voltage.  From  these  results  the  total 
losses  and  the  efficiency  at  any  load  may  be  calculated. 

A motor  running,  without  load,  or  with  part  load,  on  an  un- 
balanced circuit,  is  likely  to  absorb  widely  different  amounts 
of  power  in  its  coils ; one  coil  may  even  return  power  to  the 
circuit,  while  the  others  absorb  the  power  required  for  opera- 
tion plus  that  returned.  In  all  such  cases,  the  two-wattmeter 
method  of  measuring  the  power  gives  the  net  power  absorbed 
by  the  machine. 

4.  Power  Factor.  — The  power  factor  at  any  load  may  also 
be  calculated  from  the  circle  diagram  or  it  may  be  obtained  for 
various  loads  by  means  of  computations  made  from  the  readings 
of  amperemeters,  voltmeters,  and  wattmeters. 

5.  Regulation  and  Torque.  — The  exact  regulation  of  a ma- 
chine may  be  determined  by  actual  running  tests  under  load,  in 
which  the  actual  slip  is  measured,  but  the  percentage  slip  may 
be  taken  to  be  approximately  equal  to  the  percentage  drop  of 
voltage  in  the  armature  windings.  The  starting  torque  can  be 
measured  by  clamping  a lever  upon  the  pulley,  and  measuring 
the  pull  at  the  end  of  a fixed  length  of  arm.  For  a machine 
having  an  improperly  divided  starting  rheostat  associated  with 
the  armature,  this  gives  a value  which  is  higher  than  the  torque 
against  which  the  motor  will  start  and  run  up  to  speed.  In 
such  a machine,  the  standing  torque  and  starting  torque  are  dif- 
ferent ; but  in  a machine  having  a properly  arranged  starting 
rheostat  associated  with  the  armature,  the  maximum  standing 
torque  and  the  maximum  starting  torque  are  equal,  and  are 
practically  equal  to  the  maximum  torque  which  the  machine 
can  exert.  The  torque  and  slip  are,  of  course,  obtainable  from 
the  circle  diagram.  All  desired  information  in  regard  to  the 
operation  of  induction  motors  may  be  determined  bj?  purely 
electrical  measurements,  and  to  a high  degree  of  accuracy  for 
commercial  measurements.  Using  commercial  amperemeters, 
voltmeters,  and  wattmeters  which  have  been  properly  calibrated, 
the  errors  probably  affecting  the  full  load  efficiency  and  power 
factor  need  not  exceed  one  per  cent. 

The  operating  characteristics  of  a 50  horse  power,  four-pole, 
tri-phase,  60-cycle,  440-volt  motor  are  given  in  Fig.  488.  This 
shows  a drop  in  speed  of  about  5 per  cent  from  no  load  to  full 
load,  while  the  efficiency  rises  to  about  88  per  cent  at  slightly 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


857 


over  one  half  load.  It  is  noted  that  the  power  factor  drops 
rapidly  for  loads  smaller  than  about  one  half  load.  In  large 
machines  the  efficiency  sometimes  exceeds  95  per  cent,  while  the 


PER  CENT  OF  FULL  LOAD 

Fig.  488.  — Operating  Characteristics  of  a Tri-phase  50  H.  P.  Induction  Motor. 

slip  is  from  2 to  4 per  cent  at  full  load.  Figure  489  shows  the 
primary  current  per  phase  and  the  torque  of  a 5 horse  power, 
four-pole,  60-cycle  motor,  in  per  cent  of  their  normal  values  at 
full  load,  as  functions  of  the  armature  speed.  The  figure  there- 
fore illustrates  the  conditions  during  starting,  and  especially  the 
change  in  operating  characteristics  caused  by  change  of  voltage 
impressed  upon  the  coils  of  a phase,  obtained  by  changing  the 
connections  of  the  field  coils  from  wye  to  delta. 

Figure  490  shows  the  curves  of  torque  as  a function  of  speed 
and  of  current  when  a 25-cycle  induction  motor  is  operated  from 
various  taps  of  an  autotransformer  starter.  It  is  to  be  noted 
that  the  starting  current  is  reduced  at  serious  expense  of  start- 
ing torque,  but  that  a starting  torque  equal  to  full  load  running 
torque  may  be  obtained  with  a current  approximating  to  full 
load  current.  The  various  taps  of  the  autotransformer  give 
respectively  60,  70,  80,  and  100  per  cent  of  the  full  voltage,  and 


858 


ALTERNATING  CURRENTS 


0 10  20  30  40  60  60  70  80  90  100 

SPEED  IN  PER  CENT  OF  SYNCHRONISM 


Fig.  489.  — Curves  of  Primary  Current  and  Torque  of  a Small  Tri-phase  Induction 
Motor,  with  the  Primary  Coils  connected  in  Wye  and  in  Delta. 


PER  CENT  OF  FULL  LOAD  TORQUE 

Fig.  490.  — Torque  Speed  and  Torque  Current  Curves  of  a 25-Cycle,  Four-pole  In- 
duction Motor  when  started  and  run  on  various  Taps  of  an  Autotransformer. 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


859 


the  starting  torques  corresponding  to  the  respective  taps  are 
therefore  in  ratio  with  each  other  as  36,  49,  64,  and  100,  that 
is,  as  the  squares  of  the  respective  voltages.  The  maximum 
torques  for  the  several  voltages,  which  arise  at  55  per  cent  of 
synchronous  speed  in  this  machine,  are  in  the  same  ratio,  as 
also  are  the  torques  corresponding  to  the  several  voltages  at 
any  given  speed. 

203.  Effect  of  Frequency.  — -An  examination  of  the  formulas 
relating  to  the  design  of  induction  motors  shows  that  the  fre- 
quency of  the  current  for  which  a machine  is  designed  does  not 
greatly  affect  its  efficiency,  slip,  power  factor,  or  starting  torque, 
but  that  for  a given  speed  the  number  of  poles  selected  must  be 
directly  related  to  the  frequency.  Increasing  the  number  of 
poles  of  a given  machine  reduces  the  cross  section  of  each  pole, 
but  the  number  of  lines  of  force  at  each  pole  is  equally  reduced, 
so  that  the  magnetizing  current  is  unaltered.  Consequently, 
induction  motors  of  equal  merit  may  be  designed  for  all  reason- 
able frequencies,  though  magnetic  leakage  may  interfere  with 
the  operation  when  the  poles  become  too  numerous. 

On  the  other  hand,  when  a machine  which  has  been  designed 
for  a certain  frequency  is  operated  at  another  frequency,  the 
synchronous  speed  is  changed  in  direct  proportion  to  the  fre- 
quency, the  percentage  slip  is  practically  unaltered,  the  starting 
torque  varies  inversely  from  the  frequency,  and  the  efficiency 
and  power  factor  both  vary  in  the  same  direction  as  the  fre- 
quency because  the  magnetic  density  is  inversely  proportional 
to  the  frequency,  as  in  transformers. 

The  frequencies  which  are  commonly  used  with  induction 
motors  cover  a wide  range.  In  Europe,  50  periods  per  second 
is  frequently  adopted  for  tri-phase  and  single-phase  motors ; 
while  in  this  country  the  frequencies  of  60  periods  per  second 
and  25  periods  per  second  have  been  generally  adopted.  The 
former  is  equally  suitable  for  lighting  and  for  stationary  power 
purposes.  The  latter  was  early  adopted  at  Niagara  Falls  and  is 
used  for  plants  where  power  service  is  of  greater  importance  than 
the  lighting  service.  The  ratio  of  these  two  frequencies  being 
12  : 5 makes  frequency  changing  inconvenient  by  limiting  the 
numbers  of  pairs  of  poles  practicable  for  use  on  the  motor  and 
generator  of  the  frequency  changer,  within  a reasonable  range 
of  speeds.  The  minimum  number  of  poles  available  for  use  in 


860 


ALTERNATING  CURRENTS 


the  frequency  changing  between  these  frequencies  is  5 pairs  on 
the  25-cycle  machine  and  12  pairs  on  the  60-cycle  machine,  giv- 
ing a speed  of  300  revolutions  per  minute.  The  next  practicable 
combination  is  10  pairs  of  poles  and  24  pairs  of  poles,  giving  a 
speed  of  only  150  revolutions  per  minute,  and  therefore  making 
an  unduly  expensive  machine.  If  the  lower  frequency  is  re- 
duced to  24  periods  per  second  while  the  other  is  maintained  at 
60  periods,  rather  more  flexibility  is  obtained,  as  speeds  of  720, 
360,  240,  180,  etc.,  are  then  available  for  frequency  changers. 
Nevertheless  it  would  be  of  distinct  advantage  to  electrical  en- 
gineering in  this  country  if  the  two  commonly  used  frequencies 
were  standardized  at  values  more  readily  interconvertible. 
This  would  be  accomplished  if  the  lower  frequency  were  stand- 
ardized at  30  periods  per  second.* 

The  frequency  of  40  periods  per  second  was  adopted  in 
certain  of  the  earlier  power  transmission  plants  in  this  country, 
and  still  persists  in  a few.  In  most  instances  the  frequencies 
other  than  60  periods  per  second  and  25  periods  per  second 
have  disappeared  in  this  country. 

204.  Polyphase  Induction  Motor  with  Exciting  Current  sup- 
plied to  the  Armature.  Motor  with  Unity  Power  Factor.  — If 
a polyphase  induction  motor  has  its  armature  coils  wound  as  a 
continuous  reentrant  winding,  and  has  direct  current  supplied 
to  the  coils  through  slip  rings,  so  as  to  form  as  many  magnetic 
poles  as  there  are  poles  in  the  rotating  field  of  the  field  magnet, 
the  machine  becomes  a synchronous  motor,  and  the  lagging 
current  of  the  primary  windings  may  be  neutralized  when  the 
machine  is  running  in  synchronism,  or  a leading  current  may 
be  drawn  from  the  alternating-current  supply  mains,  f 

Now  assume  the  simple  case  of  a two-pole  tri-phase  induction 
motor  having  a continuous  reentrant  winding  tapped  to  a com- 
mutator. Suppose  that  three  brushes  bear  upon  the  commu- 
tator, spaced  120°  apart,  as  in  Fig.  491,  where  B is  the  armature 
winding,  C the  commutator,  and  d , d,  d the  brushes.  Then,  when 
the  armature  is  at  rest,  a rotating  field  may  be  created  by  current 
in  its  windings  by  attaching  the  brushes  to  three-phase  supply 
mains.  If  the  connections  are  properly  made  and  the  brushes 
are  set  at  the  proper  positions,  this  rotating  field  may  be  given 
the  same  direction  of  rotation  and  space  phase  as  the  rotating 
* Art.  183.  t Art.  168. 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


8G1 


field  set  up  by  windings  on  the  field  magnet.  If  the  two 
magnetic  fields  are  superposed  and  the  direction  of  their  fluxes 
is  the  same,  the  exciting  current  in  the  field  winding  decreases, 
since  the  total  flux  required  to  link  the  field  winding  is  limited 
by  the  amount  necessary  to  set  up  a counter-voltage  in  the 
primary  winding,  which  is  equal  to  the  vector  difference  of 
the  impressed  voltage  and  the  IR  drop  in  the  winding.  If 
the  current  introduced  through  the  brushes  is  increased  by 
raising  the  voltage  impressed  on  the  brush  circuit,  the  exciting 


Fig.  491.  — Diagram  of  a Polyphase  Induction  Motor  capable  of  High  Power  Factor. 


current  in  the  primary  winding  can  be  entirely  replaced,  and 
further  increases  of  current  through  the  brushes  tend  to  call 
for  a neutralizing  leading  current  in  the  primary  winding. 

Since  rotation  of  the  armature  does  not  affect  the  rotation 
of  the  primary  field,  and  the  commutator  maintains  a uniform 
relation  of  the  magnetic  axes  of  the  armature  winding  with 
respect  to  the  brushes,  the  same  time  and  space  relations  be- 
tween the  two  magnetic  fields  exist,  when  the  armature  is  run- 
ning at  anjr  speed,  as  exist  when  the  armature  is  standing  still. 
That  is,  whatever  may  be  the  speed  of  the  armature,  its  wind- 


862 


ALTERNATING  CURRENTS 


ing  continues  to  be  tapped  through  the  medium  of  the  brushes 
and  commutator  at  three  fixed  points  in  space.  Therefore, 
since  the  windings  spanning  between  the  brushes,  in  the  three 
segments  X , Y , and  Z , are  at  all  times  unchanged,  except  di- 
rectly under  the  brushes  as  commutation  goes  on,  the  strength 
and  rotation  of  the  field  set  up  by  the  introduction  of  the  tri- 
phase currents  through  the  brushes  is  practically  unchanged 
by  changes  in  the  speed  of  the  armature.  When  the  brushes 
are  placed  directly  under  the  field  taps  Gr , 6r,  Cr,  as  shown  in 
Fig.  491,  exciting  current  furnished  through  the  armature  will 
tend  to  lag,  and  no  advantage  will  be  obtained  in  increased 
power  factor  ; but  by  shifting  the  brushes  slightly,  and  over- 
exciting through  the  armature  brushes,  the  counter-voltage  of 
the  field  windings  may  be  made  to  take  such  a position  with 
reference  to  the  impressed  voltage  that  a leading  current  will 
be  drawn  from  the  mains  sufficient  to  neutralize  the  lagging 
current  to  the  armature,  or  even  cause  a resultant  leading  cur- 
rent in  the  supply  mains. 

By  joining  the  commutator  bars  by  resistance  A , the  motor 
may  be  run  as  an  ordinary  short-circuited  armature  induction 
motor,  while  the  exciting  current  is  being  supplied  through  the 
brushes  d,  d , d.  Evidently  in  this  case  part  of  the  current 
entering  through  the  brushes  is  wasted  in  the  parallel  path 
through  A , but  by  making  the  length  of  armature  coil  be- 
tween the  commutators  sufficiently  small,  the  resistance  of  A 
between  brushes  can  be  made  great  enough  to  reduce  the  cur- 
rent through  it  to  a reasonably  small  proportion.  This  gives 
a motor  with  good  starting  torque  and  good  regulation,  and  also 
affords  opportunity  for  control  of  the  power  factor.  Various 
modified  arrangements  for  accomplishing  the  purpose  may  be 
used  ; thus,  two  separate  windings  may  be  placed  in  the  arma- 
ture slots,  one  to  serve  purely  in  the  capacity  of  a commutated 
field  magnetizing  winding,  and  the  other  a short-circuited  wind- 
ing, which  makes  unnecessary  the  use  of  resistance  A shown  in 
the  figure.  The  plan  is  equally  applicable  to  single-phase  and 
polyphase  machines. 

After  the  motor  has  been  started  in  this  way  upon  the  high- 
starting  torque  and  power  factor,  the  brushes  can  be  raised  and 
the  commutator  can  be  short-circuited,  thus  reducing  the  ma- 
chine to  an  ordinary  induction  motor  having  all  of  the  advan- 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


863 


tages  of  low  armature  resistance,  or  it  may  be  allowed  to  run 
as  a self-regulating  motor  of  high  power  factor.  In  order  to 
start  the  motor  through  the  armature,  the  requisite  exciting 
current  can  be  obtained  by  connecting  the  brushes  d,  d,  d to 
the  supply  mains  through  the  medium  of  an  autotransformer, 
or  transformer  having  a suitable  number  of  voltage  regulation 
taps,  as  shown  at  _F,  Fig.  491.  The  effect  of  such  lagging  cur- 
rent as  flows  to  the  transformer  F can  be  neutralized  by  properly 
placing  the  brushes. 

205.  Electromagnetic  Repulsion  and  Repulsion  Motors.  — If  a 

coil  of  wire  is  held  in  an  alternating  magnetic  field  in  such  a 
way  that  the  lines  of  force  pass  through  its  turns,  an  alternating 
voltage  is  set  up  in  it  which  has  90°  difference  of  phase  from 
the  alternating  magnetic  flux.  This  in  turn  causes  a current 
in  the  coil,  and  the  coil  experiences  a force  at  each  instant  tend- 
ing to  move  it  in  the  magnetic  field,  which  is  proportional  in 
magnitude  and  direction  to  the  product  of  the  corresponding 
instantaneous  values  of  current  and  magnetic  flux,  paying  due 
attention  to  their  relative  algebraic  signs  ; and  the  average  force 
for  a cycle  of  flux  is  equal  to  the  average  of  the  instantaneous 
torques  during  the  period.  If  the  coil  could  have  no  self- 
inductance, and  the  phase  of  the  current  could  therefore  be  in 
quadrature  with  that  of  the  magnetic  flux,  the  average  forces 
during  alternate  quarter  periods  would  be  equal  but  in  opposite 
directions  (compare  Fig.  199),  and  the  average  force  during 
a whole  period  would  be  zero,  so  that  the  coil  would  have  no 
tendency  to  move  ; but  in  all  practical  cases  a coil,  or  even 
a flat  disk,  must  have  some  self-inductance,  so  that  the  current 
lags  behind  the  impressed  voltage,  and  the  current  phase  is 
therefore  more  than  90°  behind  the  phase  of  the  magnetism. 
In  this  case  the  instantaneous  values  of  the  force,  when  plotted 
in  a curve,  give  a figure  similar  to  the  dotted  curve  in  Fig.  200. 
The  ordinates  of  the  large  loops  represent  a negative  or  repul- 
sive force,  and  the  ordinates  of  the  small  loops  a positive  or 
attractive  force,  and  the  summation  of  the  instantaneous  forces 
during  a period  has  a finite  negative  value.  This  shows  that 
the  coil  experiences  a repulsive  force  which  tends  to  move  it 
out  of  the  magnetic  field.  If  the  coil  is  pivoted,  the  force  tends 
to  turn  it  into  such  a position  that  the  lines  of  force  of  the  field 
do  not  thread  through  its  turns.  The  conditions  here  set  forth 


864 


ALTERNATING  CURRENTS 


were  first  fully  explained  and  illustrated  in  a remarkable  lecture 
by  Professor  Elihu  Thomson.* 

If  an  armature  is  wound  with  uniformly  spaced  short-cir- 
cuited coils  or  conductors,  the  repulsive  effects  in  the  different 

coils  will  balance  each  other  when 
the  armature  stands  still  ; but  if 
the  coils  have  their  independent 
ends  separately  connected  to  the 
opposite  bars  of  a commutator 
having  as  many  bars  as  there  are 
sets  of  conductors  in  the  arma- 
ture, brushes  may  be  so  arranged 
as  to  short-circuit  each  coil  when 

Fig.  492.  — Diagram  for  Showing  the  ^ ^ a position  to  gi\e  a force 
Principle  of  a Repulsion  Motor  with  in  one  direction.  This  arrauge- 
Open  Circuit  Armature  Coils.  , . j , r>  r 

ment  was  suggested  by  Professor 
Thomson,!  and  is  illustrated  in  Fig.  492,  and  represents  the 
first  type  of  repulsion  motor.  The  motor  is  self-starting,  and 
runs  by  virtue  of  the  repulsion  between  the  magnetic  field  and 
the  coils,  which,  as  they  come  into  the  active  position,  are  short- 
circuited  by  the  brush  connections.  Such  a motor  is  bulky, 
inefficient,  and  expensive,  since  only  a portion  of  the  armature 
can  be  made  continuous^  effective. 

A better  method  of  arranging  a repulsion  motor  is  to  use  an 
armature  with  a re-entrant  winding  and  commutator  like  a 
direct-current  armature,  the  field  magnet  being  laminated,  as 
originally  developed  by  Anthony,  Jackson,  and  Ryan.  The  ar- 
rangement is  as  illustrated  in  Fig.  493.  In  this  figure  the  lead 
wire  cc  makes  a short-circuiting  connection  between  the  brushes 
b , b'  which  rest  on  the  commutator  to  which  the  armature  wind- 
ing is  attached.  The  brushes  are  usually  set  a little  less  than  45° 
from  the  neutral  plane.  Then  the  alternating  poles  N-S  induce, 
by  transformer  action,  currents  in  the  armature  winding  which 
is  short-circuited  through  the  brushes  and  conductor  cc  The 
voltage  induced  in  the  armature  winding  is  zero  when  meas- 
ured between  commutator' bars  lying  on  a diameter  perpendic- 
ular to  the  magnetic  axis  of  the  inducing  flux,  and  is  a maximum 
when  measured  between  commutator  bars  on  a diameter  parallel 

* Novel  Phenomena  of  Alternating  Currents,  Trans.  Amer.  Inst.  E.  E..  Vol. 
4,  p.  160.  t Trans.  Amer.  Inst  E.  E.,  Vol.  4,  p.  160. 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


865 


to  the  flux.  A large  current  flows  through  conductor  cc  when 
the  brushes  are  placed  in  the  latter  position,  but  the  currents 
flow  in  the  armature  conductors  so  that  they  neutralize  each 
other’s  torque.  When  the  brushes  are  placed  in  the  position 
at  right  angles,  no  current  flows  and  therefore  no  torque  is  pro- 
duced. Consequently,  the  brushes  are  placed  in  a compromise 
position,  giving  a considerable  current  flow  accompanied  by  the 
jv  oduction  of  a considerable  torque. 


A convenient  method  of  determining  the  action  in  the  repul- 
sion motor  is  to  resolve  the  magneto-motive  force  OP  of  the 
field  magnet  into  two  quadrature  components,  one  of  which, 
OM,  is  in  line  with  the  short-circuited  brushes,  and  the  other, 
OF,  at  right  angles  thereto.  The  field  windings  are  wound 
through  the  holes  or  slots  in  the  outer  ring  of  Fig.  493,  and 
produce  the  magneto-motive  force  OP,  proportioned  to  their 
ampere-turns.  Now,  since  the  vector  expression  OP=  OF  + OM 
holds  true,  we  can  replace  these  field  windings  by  two  sets  in 
space  quadrature,  but  connected  electrically  in  series,  such  that 
one  set,  which  we  shall  call  the  torque  or  F coils,  creates  the 
magneto-motiye  force  OF,  and  the  other,  which  we  shall  call 
the  mutual  or  M coils,  creates  the  magneto-motive  force  OM. 


8G6 


ALTERNATING  CURRENTS 


Assume  first  an  ideal  motor  in  which  the  F and  M coils  and 
the  armature  winding  are  of  negligible  resistance.  Remember 
that  the  armature  winding  is  of  the  closed  coil  variety  and  is 
short-circuited  between  the  brushes  by  wire  cc.  Assume  also 
that  the  reluctance  of  the  magnetic  circuit  is  uniform  and  the 
magnetic  leakage  zero.  If  the  armature  is  stationary  and  an 
alternating  voltage  E is  impressed  across  the  extremities  of  the 
F and  M coil  circuit,  causing  a current  I to  flow  through  the 
circuit,  an  alternating  magneto-motive  force  is  set  up  along 
the  line  MM\  and  another  along  the  line  FF'  by  reason  of  the 
M and  F windings  respectively.  The  flux  from  the  M wind- 
ings or  poles  induces  in  the  armature  windings  bands  of  cur- 
rents which  complete  their  circuit  from  b through  c to  b\  The 
armature  windings  having  negligible  resistance  and  leakage  in- 
ductance by  assumption,  this  armature  current  IM  flows  with 

negligible  voltage  drop.  The  value  of  this  current  is  Im  = ^Z 

8 

where  s"  is  the  ratio  of  transformation  of  the  transformer  thus 
formed  between  the  M coils  and  the  armature  windings  or  A 

coils,  or  s"  = — , where  nA  and  nM  are  the  equivalent  numbers 
nM 

of  turns  in  the  A and  M coils  respectively.  The  resistance  of 
the  M coils  being  also  assumed  to  be  negligible,  IM  in  the  A 
coils  acts  to  destroy  the  mutual  reactance  of  the  M coils,  like 
the  current  in  the  secondary  circuit  of  any  transformer;  then, 
there  being  no  induced  voltage  drop  in  the  A coils,  the  induced 
voltage  in  the  M coils  is  zero,  which  means  that  the  magneto- 
motive force  of  the  M coils  has  been  counterbalanced  and 
neutralized  by  that  of  the  A coils.  On  the  other  hand,  the 
magneto-motive  force  caused  by  the  current  I in  the  F coils 
does  not  set  up  current  in  the  armature  or  A coils,  because  the 
brushes  b and  b'  rest  on  neutral  points  with  reference  to  the 
voltages  induced  by  reason  of  the  magnetic  flux  along  FF' 
caused  by  this  F magneto-motive  force.  Hence,  there  must  be 
an  induced  counter-voltage  in  the  F coils.  This  is  EF  = — IXF, 
where  I is  the  line  current  and  XF  the  reactance  of  the  F coils. 
The  impressed  voltage  E is  thus  all  used  in  drop  through  the  F 
coils,  in  the  ideal  motor,  there  being  no  drop  in  the  M coils,  the 
other  part  of  the  complete  field  circuit,  since  their  reactance  is 
destroyed  by  the  mutual  effect  of  the  A coils,  and  all  resistance 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


867 


is  considered  negligible.  The  voltage  EM,  across  the  M coils, 
cannot  be  absolutely  equal  to  zero,  or  EF  be  equal  to  — IXF,  in 
an  actual  machine,  since  neither  resistance  nor  leakage  reactance 
can  be  entirely  eliminated,  but  for  our  purposes  of  discussion 
the  assumption  of  an  ideal  motor  is  warranted. 

The  foregoing  two  effects  are  combined  in  the  armature  and 
an  alternating  magnetic  flux  <3?^,  emanates  from  the  field  mag- 
net on  the  line  FF and  heavy  sheets  of  alternating  currents  Im 
flow  through  the  armature  coils  and  the  conductor  cc  between  b 
and  b' . Moreover,  the  time  phases  of  <3>^  and  IM  are  in  opposi- 
tion, since  IM  must  be  in  phase  opposition  to  the  current  / in  the 
inducing  coils  to  accord  with  the  laws  of  the  transformer  if 
losses  and  magnetic  leakage  are  negligible.*  Therefore,  the 
armature  current  IM  and  field  or  F flux  <3?^  are  in  phase  relation 
and  space  position  to  exert  a torque  upon  one  another.  Now 
suppose  that  the  armature  rotates  under  the  influence  of  this 
torque.  Then  the  two  sheets  of  cond  uctors  lying  on  the  armature 
between  b and  b'  will  generate  a voltage  Ec  between  those  points, 
due  to  cutting  the  flux  $ F,  so  that  Ec  = kS<& F,  where  Jc  is  a con- 
stant, and  S is  the  speed  of  rotation  in  per  cent  of  the  frequency 
/.  But  <3?^  is  in  phase  with  I.  Idence  Ec  and  I are  in  the  same 
time  phase  and  Ec  is  in  phase  opposition  with  IM.  The  voltage  Ec 
is  alternating,  because  its  instantaneous  values  are  proportional 
to  <f)F  at  each  instant,  but  it  is  also  proportional  to  the  speed  of 
the  armature.  The  current  component  Ic  flowing  through  the 
armature  under  the  influence  of  Ec  creates  a magnetic  flux  <3>c 
which  passes  through  the  M coils  and  sets  up  in  them  a voltage 
Eu  lagging  90°  behind  it  and  proportional  to  its  rate  of  change. 
E 

But  Ic  = —A.,  when  XM  is  the  reactance  (mutual)  offered  on 

Xju 

account  of  the  flux  <3>c,  and,  therefore,  since  the  circuit  is  as- 
sumed to  be  purely  reactive,  Ic  lags  90°  behind  Ec , and  hence 
is  270°  behind  or  90°  ahead  of  IM.  This  makes  EM  ninety  de- 
grees behind  EF  = — IX F in  time  phase.  But  since  EF  is  90° 
behind  /,  EM  must  be  in  phase  opposition  to  I and  the  vector 
product  of  the  two  must  represent  real  power. 

We  now  have  two  induced  voltages  EM  and  EF  in  quadra- 
ture which  cause  the  drop  E in  the  field  circuit  or 

E = -E'=-  VEm*  + E/. 

* Chap.  X. 


8G8 


ALTERNATING  CURRENTS 


Likewise,  we  have  two  quadrature  components  of  current  h 
the  armature  IM  and  /c,  which  are  flowing  in  what  is  the  equiv 
alent  of  a transformer  secondary  to  the  M windings,  and  the 
current  through  the  brushes  must  be  equal  to  their  resultant,  or. 

IA  = VIJ  + I*. 

The  stator  current  is 

j- E F E cos  A 

F 

where  A is  of  such  value  that  tan  A = — Evidently  EF,  EM, 

EF 

and  I can  be  expressed  graphically  by  means  of  locus  diagrams. 
Thus,  let  OE , Fig.  494,  be  the  impressed  voltage.  Then  since 


Fig.  494. — Vector  Diagram  of  Relations  in  a Repulsion  Motor  having  Negligible 
Resistance  and  Leakage  Reactance. 


the  induced  voltages  Eu  and  EF  are  in  quadrature  and  their 
vector  sum  equals  — OE , they  may  be  represented  respectivel}'  by 
the  vectors  EQ  and  QO.  The  point  Q travels  from  E at  stand- 
still, when  Em  is  zero,  to  0 , when  EF  is  zero,  along  the  semi- 
circle OQE  The  current,  I,  in  the  stator  is  at  right  angles  to 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


869 


OQ  - — Er,  under  the  conditions  assumed.  It  may  therefore 
be  i presented  by  the  line  01  for  the  particular  value  of  Ep 
slum  in  the  figure.  The  point  I travels  from  X when  the 

_ Tl 

armcure  is  at  rest  and  the  current  I has  the  value  — , to  0 

XF 

whe  infinite  speed  has  been  reached  and  EF  is  zero.  The  sec- 
ondly current  is  composed  of  the  two  components  IM  and  /e. 
As  hs  been  explained,  1m  sets  up  a magneto-motive  force  equal 
and  pposite  to  that  of  I in  the  M coils.  It  may  therefore  be 
represented  by  the  line  OJ.  The  point  J travels  along  the  cir- 
cula  locus  OJU  with  change  in  load,  but  alwaj^s  takes  such  a 
positon  that  JOI  is  a straight  line.  The  scalar  value  of  IM  is 

equi  to  I — , where  s equals  the  ratio  between  the  turns  in  the 
8 

M fild  coils  and  the  F field  coils,  or  — , and  s'  is  the  ratio  be- 

nF 

tweo  the  effective  armature  turns  and  the  F turns,  or  — . This 

nF 

is  erdent,  since  I and  the  mutually  induced  armature  current 

Iu  rust  have  the  ordinary  transformer  relation  = nM=  A = 

I nA  s s" 

Thi  ratio  is  constant  so  long  as  the  brushes  are  not  moved,  so  that 
if  I races  a semicircle,  J must  follow  a similar  locus.  The  di- 


ame^r  OU  equals  OX  — . The  second  component  of  armature 
s 

curint  /c,  as  can  be  gleaned  from  the  preceding  discussion,  is  al- 
way  in  phase  with  EF  and  at  right  angles  to  Ec.  It  is  also  pro- 

_ JT 

porDnal  to  I and  the  armature  speed.  Since  Ic  = — £ , J=  — — 

XA  XF 

and  he  reactances  are  proportional  to  the  square  of  the  numbers 
of  trns  in  which  the  voltages  Ec  and  EF  are  induced,  then 

j n 2 

-S=~  A’,  and,  as  will  be  shown  later,  Ec=  EFs'  S ; hence 
I ™FnA 

s 

Ic  = — , S being  the  armature  speed  in  per  cent  of  the  impressed 

freqency.  Being  proportional  to  and  set  up  by  Ec , Ic  may 
be  i presented  by  the  line  OK.  The  point  K travels  the  semi- 

E 

circ;  OKV \ which  has  a diameter  OV  equal  to  — f-,  and  reaches 

Ka 

V a infinite  speed  of  the  armature.  The  resultant  of  OJ  and 


870 


ALTERNATING  CURRENTS 


OK  is  the  total  armature  current,  and  it  also,  as  seen  by  the 
geometrical  construction,  traces  a circular  locus  having  a diam- 
eter uv. 

The  power  input  is  01  x OE  cos  E 01  = IE  cos  6 — EUI  and 
may  be  represented  by  IW  in  the  figure.  The  torque,  under 

E I 

the  assumption  made,  is  . Since  the  generated  voltage  EM 


is  proportional  to  the  product  S <fv,  and  d> F is  the  maximum 
value  of  the  flux  from  the  E coils,  and  since  <t>F  is  proportional 

to  Ef,  we  have  S x x tan  (90°  - 6)  yo  Zg.wer  factor . 1 1 mav, 

EF  y J cos(90°  — d)  y' 

therefore,  be  represented  by  the  line  TE.  The  output  under 

the  assumption  of  negligible  losses  is  equal  to  the  input.  The 

E E 

angle  of  lag  6 equals  cos^1— tan-1— £ • 

Ef  Em 

The  relation  of  the  voltages  in  the  two  sets  of  imaginary  field 

coils  Em  = EpSs  may  be  found  in  this  manner:  — = ^ S = s'* S'. 

Ef  nF 

E 7t 

and  — 5 = — = — = s",  from  which  EM  = EFSs.  Substituting  this 
EM  s 


in  the  formula  E—  — V E^1  + EF2  gives  EF  = — 
E 


Also 


1=  — --  = 


XF  Xps/l  + Sh  2 


VI  + w 

Substituting  the  values  of  Ic  and  IM, 


obtained  in  terms  of  /,  *S',  s,  and  s',  in  the  formula  IA=V  I2  + IM2 


gives 


Ia  = 


— Vs2  + *S'2,  and  therefore 
s' 


Ia  = 


_Ej:_ 

A>' 


Vs2  + S2  — 


EVs2  + S2 
XpsW  lTw' 


From  the  above  formulas  and  the  diagram  much  information 
concerning  the  operation  of  the  repulsion  motor  may  be  ob- 
tained. If  the  speed  is  such  that  S — 1,  which  is  sometimes 
called  synchronous  speed,  since  then  360  electrical  degrees  of 
the  polar  pitch  is  turned  through  by  the  motor  armature  dur- 
ing the  time  of  one  cycle  of  the  impressed  voltage,  IA  reduces 


to 


E 


When  s is  also  made  unity  by  placing  the  brushes  45° 


from  the  neutral  plane,  the  primary  current  becomes  I = — — 


E 


V2AV 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


871 


By  completing  the  circular  arcs  in  the  locus  diagram  the  loci 
are  obtained  for  the  machine  driven  backward,  in  which  case  it 
runs  as  a generator.  The  conditions  are  illustrated  by  the 
dotted  halves  of  the  circular  loci  of  Fig.  494. 


Fig.  495. — Vector  Diagram  of  a Repulsion  Machine  as  Generator  and  Motor  when 
Winding  Resistance  and  Reactance  are  Neglected. 

Figure  496  shows  diagrammatically  the  windings  of  a repul- 
sion motor  when  the  stator  coils  are  actually  divided  into  parts. 
In  this  case  the  brushes  are  directly  under  the  windings  marked 
“ mutual  coils.”  At  standstill  the  magneto-motive  forces  of  the 
armature  and  mutual  field  currents  neutralize  each  other  except 
for  the  drops  required  to  overcome  resistance  and  leakage  re- 
actance. The  sheets  of  current  in  the  armature  react  upon  the 
unneutralized  flux  which  is  set  up  by  the  current  in  the  “ torque 
coils,”  causing  the  torque  which  results  in  rotation.  As  in  the 
discussion  of  the  simple  repulsion  motor,  the  voltage  set  up  in 
the  armature  conductors  by  the  rotation  of  the  armature  is 
(neglecting  the  iron  loss  angle)  in  phase  with  the  torque  flux 
and  hence  with  the  field  current.  The  component  of  armature 
current  caused  by  this  flux  lags  nearly  90°  behind  this  voltage, 


872 


ALTERNATING  CURRENTS 


and  hence  its  flux  entering  the  mutual  coils  sets  up  a voltage  in 
those  coils  in  quadrature  with  the  voltage  in  the  torque  coils. 
An  extra  current  must  flow  through  the  stator  circuit  to  oppose 
the  magneto-motive  force  of  this  new  component  of  the  armature 
current.  At  synchronous  speed  the  two  stator  fields  become 
equal,  and  beyond  that  speed  the  mutual  field  becomes  the 
stronger,  for  which  reason  the  repulsion  motor  commutates 
better  below  synchronous  speed  than  above.  The  formulas  and 
diagram  given  for  the  simple  repulsion  motor  apply  also  to  this 
modified  type. 


As  explained,  the  usual  forms  of  commercial  repulsion  motor 
have  field  cores  like  those  of  induction  motors.  In  order  then 
to  prevent  undue  reactance  in  the  armature  circuit,  caused  by 
the  low  reluctance  of  the  iron  path,  special  arrangements  may 
be  used  as  shown  in  Fig.  497.  In  this  case,  in  a two-pole 
machine,  the  pair  of  mutually  induced  poles  are  at  M and 
while  the  torque  poles  excited  by  current  in  the  armature  are  at 
F and  F' . The  two  short-circuited  brushes  are  directly  under 
the  points  M and  M'  and  collect  the  mutually  induced  currents 
in  the  armature,  neutralizing  the  reactance  in  the  field.  The 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


873 


two  brushes  under  F and  F'  are  connected  in  series  with  the 
field  coils  and  the  mains.  The  armature  currents  in  the  short 
circuit  paths  react  on  the  torque  poles  F and  F'  and  cause  the 
armature  to  rotate.  The  series  current  through  the  rotor  sets 
up  the  motor  flux,  which  produces  a torque  by  its  action  on  the 
current  in  the  stator  coils.  The  slxort-circuited  induced  rotor 


Fig.  41)7.  — Compensated  Repulsion  Motor,  Compensating  Current  in  Armature. 


current  opposes  its  magneto-motive  force  to  that  of  the  cur- 
rent in  the  stator  coils,  reducing  the  reactance  of  the  latter  to 
a reasonable  value.  At  starting,  the  stator  circuit  has  very 
low  impedance.  When  the  machine  is  running,  however,  the 
top  and  bottom  bands  of  the  rotor  conductors  cut  the  flux 
set  up  vertically  by  the  line  current  in  the  rotor.  This  creates 
a voltage  in  phase  with  the  line  current,  which  in  turn  sends, 
through  the  short-circuited  rotor  circuits,  a current  lagging 
nearly  90°  behind  the  line  current.  The  flux  from  this  current 
passing  into  the  stator  at  M or  M'  creates  in  the  stator  coils  an 
induced  voltage  lagging  approximately  90°  behind  its  own 
phase,  and  hence  in  approximate  opposition  to  the  line  current. 
Therefore,  the  stator  coils  have  set  up  in  them  a counter-vol- 


874 


ALTERNATING  CURRENTS 


tage  proportional  to  the  speed  of  the  rotor,  which  is  similar  in 
effect  to  the  counter-voltage  in  the  armature  of  a direct  or 
alternating-current  series  motor.  In  this  motor  the  excitation 
of  the  field  magnet  is  furnished  by  currents  in  the  armature 
conductors,  which  are  at  full  line  frequency  when  the  rotor  is 
standing  still  ; at  synchronism  the  frequency  in  each  conductor 
is  fcero ; and  above  line  synchronism  the  conductor  frequency 
again  gains  a value  proportionate  to  the  relative  speed  above 
synchronism  of  the  rotor  with  respect  to  the  frequency  of  the 
line  current.  As  a result,  at  synchronism  the  power  factor  ap- 
proaches unity,  and  above  synchronism  the  motor  may  draw  lead- 
ing current  from  the 
line. 

Figure  498  shows 
a cross  section  of  a 
single-phase  motor 
which  starts  as  a re- 
pulsion motor  and 
runs  as  an  induction 
motor.  In  this  case 
the  brushes  are  mov- 
able, and  when  the 
motor  comes  to  speed 
the  commutator  is 
short-circuited  by  a 
ring  of  copper  and 
the  brushes  are  raised  from  contact.  The  movements  are 
accomplished  by  means  of  a centrifugal  governor  on  the  shaft 
within  the  armature  spider. 

Figures  499  and  500  show  the  normal  performance  of  a 20 
horse  power,  60-cycle,  six-pole  motor  of  the  type  illustrated  in 
Fig.  498.  As  the  repulsion  motor  commutates  best  at  speeds 
below  the  theoretical  synchronous  speed  for  its  armature  as  an 
induction  machine,  this  combination  of  repulsion  motor  starting 
and  induction  motor  running  gives  an  excellent  form  of  single- 
phase machine. 

206.  Series  Alternating  Current  Motors.  — Since  a direct- 
current  series  motor  does  not  reverse  its  direction  of  rotation 
when  the  current  is  simultaneously  reversed  in  its  field  and 
armature  windings,  it  might  be  expected  to  run  when  supplied 


Fig.  498. — Cross  Section  of  a Single-phase  Motor  con- 
structed to  start  as  a Repulsion  Motor  and  run  as  an 
Induction  Motor. 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


875 


with  an  alternating  current.  This  is  the  case  when  the  field 
and  armature  cores  are  sufficiently  well  laminated  to  avoid  ex- 


Fig.  499.  — Torque  and  Current  Curves  of  Single-phase  Induction  Motor  arranged  to 
start  as  a Repulsion  Motor,  such  as  is  shown  in  Fig.  498. 

cessive  eddy  currents  and  the  electric  and  magnetic  circuits  are 
so  designed  as  to  keep  self-inductive  reactance  down  to  reason- 
able limits. 

In  the  case  of  the  direct-current  motor  it  is  common  to  make 
the  fields  strong  to  prevent  skewing  of  the  flux,  with  resultant 


PER  CENT  OF  FULL  LOAD  HJ3. 

Fig.  500.  — Performance  Curves  of  Motor  such  as  is  shown  in  Fig.  498. 


87G 


ALTERNATING  CURRENTS 


poor  commutation,  while  the  armature  is  made  magnetically 
weak  for  the  same  purpose,  and  in  order  to  reduce  the  self- 
inductance of  the  coils  under  commutation  to  a minimum.  In 
the  alternating-current  series  motor  the  field  magnet  is  made 
magnetically  weak  to  reduce  its  self-inductance  to  a minimum, 
while  the  armature  is  made  relatively  strong  as  a magnet.  The 
self-inductance  of  the  latter  is  reduced  or  overcome  by  means  of 
Compensating  coils  as  described  below. 

The  general  construction  of  the  motor,  as  frequently  used,  is 
seen  in  Fig.  501.  In  this  figure  the  field  coils  A are  undistrib- 


Fig.  501.  — Diagrammatic  Sketch  of  a Series  Alternating  Current  Motor. 


uted  and  surround  the  armature,  the  armature  winding  B is  of 
the  continuous  closed  circuit  type,  and  the  compensating  coils  C 
are  distributed  along  the  long  pole  faces  and  carry  a current  which 
opposes  the  magneto-motive  force  of  the  armature  current. 
Sometimes  the  field  windings  are  distributed,  but  this  is  more 
or  less  disadvantageous,  as  it  interferes  with  the  best  use  of  the 
compensating  windings.  More  often  the  ordinary  polar  projec- 
tions are  used  with  the  compensating  windings  distributed  over 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


877 


Fig.  502. 


-Diagram  of  an  Inductively  Compensated  Series 
Motor. 


the  pole  shoes.  For  mechanical  convenience  every  other  pole 
only  need  bear  a field  winding,  the  alternate  ones  merely 
carrying  the  magnetic  fiux. 

The  methods  of  connecting  up  the  windings  are  shown  dia- 
grammatically  in 
Figs.  502  to  507. 

In  Fig.  502  is 
shown  an  Induc- 
tively compensated 
series  motor.  In 
this  case  the  com- 
pensation winding 
distributed  over 
the  pole  faces  is 
short-circuited 
upon  itself,  and 
by  the  effect  of 
mutual  induction 
it  has  generated 
within  it  a short-circuited  current  having  approximately  the 
same  magneto-motive  force  as  that  in  the  armature.  In  this 
way  the  armature  self-inductance  is  in  large  part  destroyed. 

The  armature  may 
be  considered 
equivalent  to  the 
primary  winding 
and  the  compensa- 
tion winding  to  the 
secondary  winding 
of  a transformer, 
and  the  transformer 
diagrams  previously 
given  apply  to  this 
case.* 

Figure  508  shows 

Fig.  503.  - Conductively  Compensated  Series  Motor.  the  connections  0f  a 

Conductively  compensated  series  motor.  Here  the  compensation 
winding  C is  in  series  with  the  field  winding  A and  the  arma- 
ture B . By  properly  proportioning  the  turns  in  winding  O the 

* Art.  124. 


878 


ALTERNATING  CURRENTS 


motor  may  be  Overcompensated  or  Undercompensated,  i.e.  C may 
be  made  magnetically  stronger  or  weaker  than  B.  It  will  be 
noted  that  less  voltage  will  be  available  for  the  armature  with 

this  connection 


than  when  the  in- 
ductively compen- 
sated arrangement 
is  used. 

Figure  504  shows 
the  connections 
when  the  field  and 
compensation  wind- 
ings are  connected 
in  series  with  each 
other  and  the  req- 
uisite current  is 
induced 
in  them  by  the  ar- 
mature acting  as  a transformer  primary  winding.  In  this  case 
full  line  voltage  is  available  across  the  armature  brushes. 

Figure  505  shows  the  field  windings  A in  series  with  the 
compensation  windings  C arranged  for  electrical  connection 


Fig.  504.  — Series  Motor  with  Inductively  Connected  Field  _ ^ nail  v 

and  Compensation  Windings.  J 


Fig.  505. — Series  Repulsion  Motor.  Field  and  Compensation  Windings  Independent 
of  the  Armature  Winding. 


with  the  supply  circuit  independently  of  the  armature  wind- 
ings B.  With  this  arrangement  the  stator  and  rotor  cir- 
cuit voltages  can  be  controlled  independently  of  each  other. 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


879 


A motor  so  con- 
nected is  sometimes 
called  a Series  re- 
pulsion motor. 

An  arrangement 
somewhat  similar  to 
the  last  is  shown  in 
Fig.  506,  where  the 
compensating  coils 
alone  are  connected 
independently  to  the 
line,  and  hence  the 
value  of  the  com- 
pensation Can  be  Fig.  506. — Series  Repulsion  Motor.  The  Compensation 
varied  at  will  with-  Winding  is  connected  to  the  Line  independently. 

out  interfering  with  the  voltage  applied  to  the  armature  and 
field  windings,  which  are  in  series. 

Figure  507  diagrams  a true  repulsion  motor,  similar  to  that 
shown  in  Fig.  496,  except  that  the  current  of  the  stator  coils 

setting  up  the  torque 
flux  is  obtained  directly 
from  the  armature. 

Figure  508  shows  a 
conductively  compen- 
sated series  motor  such 
as  is  illustrated  in  Fig. 
503,  except  that  a non- 
reactive shunt  D is 
added  around  the  field 
windings  for  the  pur- 
pose of  reducing  the 
impedance  of  the  field 
circuit. 

It  is  evident  that  by 
bringing  the  terminals 
of  the  field  winding  A,  the  armature  terminals  J3,  and  the  ter- 
minals of  the  compensation  winding  C to  a proper  switching 
device  a motor  can  readily  be  given  any  of  the  connections 
shown  in  Figs.  496,  and  502  to  507.  In  this  way  a motor 
may  be  started  with  the  connection  that  gives  best  commuta- 


Fig.  507. — Repulsion  Motor  with  Torque  Coils  in 
Series  with  Armature  Circuit. 


880 


ALTERNATING  CURRENTS 


tion,  power  factor,  and  torque,  and  may  then  be  converted  to 
such  other  connections  as  will  give  the  proper  speed  and  best 
power  factor  and  commutation  for  the  speed  and  load  which 
are  to  be  maintained. 

The  compensation  winding  cannot  absolutely  neutralize  the 
effect  of  the  armature  magneto-motive  force  because  of  the  mag- 
netic leakage  around  the  conductors  of  both  the  armature  and 
compensation  coils,  which  is  not  mutual,  but  bridges  across  the 

core  teeth.  Nor  will  it  fully  op- 
pose the  armature  magneto-motive 
C force  in  the  mutual  circuit  unless 
distributed  entirely  around  the  ar- 
mature when  the  armature  wind- 
ing pitch  is  180  electrical  degrees. 
Thus,  take  the  inductively  con- 
nected compensation  when  the 
Fig.  508.— Compensated  Series  Motor  compensating  current  times  the 

with  Non-reactive  Shunt  around  ratio  of  the  compensating  turns  to 
Field  Windings.  . . ° , 

the  armature  turns  is  approximately 

equal  to  the  armature  current,  which  gives  equal  and  opposite 
magneto-motive  forces  for  the  two.  Then  the  distribution  of 
magneto-motive  force  in  the  compensating  coils  will  be  different 
from  that  in  the  armature  if  the  former  is  wound  upon  an  arc  equal 
to  the  pole  width  while  the  latter  covers  180  electrical  degrees. 
If  the  armature  winding  has  a pitch  equal  to  the  width  of  the 
pole  face,  the  two  magneto-motive  forces  can  be  made  practi- 
cally equal  and  opposite  in  their  effects,  and  thejr  therefore  prac- 
tically neutralize  each  other.  Under  ordinary  construction, 
then,  the  armature  and  compensation  winding  self-inductance  is 
not  absolutely  overcome,  but  it  is  reduced  to  reasonable  values. 
When  the  motor  is  to  be  used  alternatively  with  direct  currents 
and  alternating  currents,  conductive  compensation  should  be 
used,  since  otherwise  with  the  magnetically  strong  armature 
and  weak  field,  excessive  skewing  of  the  field  flux  would  occur 
with  the  direct  currents,  with  resultant  bad  commutation  and 
low  maximum  power. 

The  alternating-current  commutator  motor  if  sufficiently  small 
may  be  started  by  throwing  it  directly  upon  the  line.  In  the 
case  of  heavy  motors,  such  as  are  used  in  railway  work,  it  is 
desirable  to  start  by  varying  the  supply  voltage  by  means  of 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


881 


transformers  or  autotransformers  having  variable  voltage  taps 
brought  out  from  the  secondary  windings.  When  necessary  to 
improve  the  power  factor,  the  field  coil  may  be  shunted  by  a 
non-inductive  resistance  as  shown  at  D in  Fig.  508. 

207.  Vector  Diagrams  of  Alternating  Current  Series  Motors, 
and  Expressions  for  Voltage.  — The  vector  diagram  of  the  series 
motor  may  be  constructed  as  in  Fig.  509.  Consider  the  con- 
ductively  compensated  type  first.  Here  the  current  flowing  is 


represented  by  OA.  The  magnetic  field  flux,  represented  by 
OM,  lags  behind  this  because  of  the  iron  loss  angle  and  by  reason 
of  the  fact  that  there  is  a considerable  amount  of  true  power 
consumed  in  the  short-circuited  armature  coils  lying  under  the 
brushes  during  commutation.  The  voltage  drop  in  the  leakage 
reactance  of  all  the  windings  is  represented  by  OC,  which  is  in 
quadrature  with  OA,  and  in  their  combined  resistance  by  OD 
which  is  in  phase  with  OA.  Thus  OF  may  be  considered  as 
the  voltage  drop  in  the  local  impedance.  The  line  FG  repre- 
sents the  voltage  drop  equal  and  opposite  to  OG',  the  voltage 
generated  in  the  field  windings  by  the  alternation  of  the  field 
flux,  OM=  <3>,  through  the  field  windings.  The  line  OB  is  the 
drop  of  impressed  voltage  equal  and  opposite  to  the  voltage  OB' 
created  by  the  armature  conductors  cutting  the  field  flux  <3?  as 
the  armature  rotates.  Voltage  OB'  must  be  in  phase  opposition 


882 


ALTERNATING  CURRENTS 


to  the  flux  OM,  as  it  tends  to  drive  a current  in  opposition  to 
the  current  caused  to  flow  by  the  impressed  voltage.  Combin- 
ing these  three  voltage  drops  OF , FGr , and  OB , gives  OJ  as 
the  impressed  voltage  for  the  current  OA  when  the  speed  is 
proportional  to  the  voltage  OB.  The  angle  of  lag  is  JO  A,  the 
power  input  is  OJ  x OA  x cos  6,  and  the  torque  is  proportional 
to  OM  x OA  x cos  B. 

Suppose  now  that  the  current  OA , Fig.  509,  is  kept  constant 
in  value  as  the  speed  varies,  a condition  entirely  possible  in 
traction  work.  Then  T>  = OM  is  constant ; and  hence  the  vol- 
tage OGr  is  constant.  Now,  suppose  the  impressed  voltage  is 
reduced  by  means  of  a controller  until  the  speed  has  lowered  to 
such  a value  that  the  induced  voltage  caused  by  the  armature  ro- 
tation is  — 0B1 ; then  the  impressed  voltage  is  OJv  the  angle  of 
lag  is  increased,  and  the  power  input  and  output  have  decreased. 
The  torque,  however,  which  is  proportional  to  the  current  OA 
times  the  flux  fl>(=  OM)  times  cos  B,  has  remained  constant. 
At  standstill  the  impressed  voltage  that  maintains  a current  OA 
is  voltage  OGr.  By  the  construction  it  is  seen  that  the  straight 
line  G-J1JJ2  extended  to  the  right  is  the  locus  of  the  impressed 
voltage  vector  for  constant  current  and  torque  but  variable 
speed.  If  the  speed  increases  beyond  the  original  value, 
considered,  for  instance,  so  that  the  armature  counter-voltage 
becomes  — OBv  the  impressed  voltage  must  become  OJ2  to 
maintain  the  same  current,  and  the  angle  of  lag  is  decreased 
and  the  input  and  output  are  increased.  It  will  be  observed 
that  the  speed  is  proportional  to  the  voltage  on  the  line  OB. 

By  considering  the  vertical  component  of  OGr  constant  and 
the  angle  S negligible  in  Fig.  509,  a simple  circular  locus  dia- 
gram can  be  constructed  for  the  compensated  series  motor  when 
operating  at  various  loads  (variable  currents)  under  constant 
voltage.  The  counter-voltage  E c generated  by  the  armature 
conductors  cutting  the  field  flux  may  be  considered  as  replace- 

E 

able  by  a variable  resistance  such  that  Rc  = —^-.  Then  the 


current  flowing 


through  the  motor  is  / = 


E 

vl2+(«t  rS2' 


where  X is  the  uncompensated  reactance,  R is  the  true  resist- 
ance of  the  windings  plus  such  additional  amount  as  is  required 
to  be  substituted  for  iron  and  commutator  losses  to  give  an  equiv- 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


883 


alent  power  loss,  and  Rc  is  the  apparent  resistance  due  to  the 
counter-voltage.  But  the  reactance  is  constant,  while  the  resist- 
ance (i2  + Rc')  varies  when  the  current,  and  hence  the  speed,  and 
with  it  Ec , varies.  Therefore,  we  have  a case  of  a circuit  contain- 
ing a constant  reactance  and  variable  resistance.*  Then  lay  off 
OR,  the  voltage  E,  Fig.  510,  and  at  right  angles  with  this  lay 

E 

off  OA  such  that  OA  = ~T.  Then  the  semicircle  OQA  repre- 


sents the  required  locus.  At  standstill  Rc  is  zero  and  the  total 

V 


Fig.  510.  — Current  Locus  of  a Series  Alternating  Current  Motor. 


resistance  is  R.  If,  then,  OQx  is  the  starting  current,  QXTX  rep- 
resents the  standstill  losses.  The  power  input  for  any  current 
OQ  is  _Pj=  El  cos  6 = OE  x QT , which  varies  with  QT.  The 

angle  of  lag  is  EOQ  = 0 ; and  tan  6—  ^-~  = where  Ea  is  the 

QT  Ea 

active 'and  Ex  the  quadrature  component  of  the  impressed  vol- 
tage. Therefore  tan  (90°  — 6)  = — = . But  X is  con- 

Er  X 


* Art.  TO  (a). 


884 


ALTERNATING  CURRENTS 


stant  and  R is  constant,  but  Rc  varies  as  the  speed  and  the 
counter-voltage  Ec.  Therefore,  a vertical  line  such  as  R V, 
Fig.  510,  can  be  erected  and  the  speed  and  counter-voltage 
will  be  approximately  proportional  to  the  intercept  WV  when 
current  OQ  flows.  WV  at  proper  scale  times  the  current  then 
equals  the  motor  output.  The  torque  is  approximately  propor- 
tional to  the  current  squared,  under  the  conditions  assumed.  By 
mean  proportionals,  OQ x2  : OQ 2 : : 0T1  : OT ; hence  the  torque 
can  be  measured  off  directly  on  the  line  OA  if  the  proper  scale  is 
used.  As  the  copper  losses  vary  as  T2,  it  is  sometimes  assumed 
that  all  losses  vary  as  P.  In  this  case  TU  would  be  propor- 
tional to  the  losses  for  current  OQ , and  UQ  be  proportional  to  the 


Field  Circuit. 

The  power  factor  of  the  motor  may  be  increased  if  desired  by 
shunting  the  field  circuit  as  shown  in  Fig.  508  at  D.  In  this 
case,  the  diagram  of  voltages  becomes  as  shown  in  Fig.  511. 
The  current  entering  the  motor  terminals  is  OA.  It  divides 
into  two  components  assumed  to  be  at  right  angles,  through  the 
non-inductive  shunt  and  the  inductive  field  coils  ; let  the  former 
component  be  OQ  and  the  latter  OR.  The  magnetic  field  flux 
OM=  <I>  is  (neglecting  the  iron  loss  angle)  in  phase  with  OP,  the 
current  in  the  field  coil ; and  the  countei’-voltage  OB'  is  in  oppo- 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


885 


sition  to  <f>  ; while  the  field-induced  voltage  GF  is  at  right  angles 
with  <f>.  Joining  the  local  impedance  drop  OF  with  FG  and 
OB  gives  an  impressed  voltage  OJ.  The  voltage  locus  is  GJ 
for  constant  current  OA  and  variable  speed.  This  arrange- 
ment makes  it  possible  by  using  a variable  shunt  resistance  to 
neutralize  the  lagging  currrent  and  even  draw  leading  current 
from  the  line.  Various  other  methods  of  controlling  the  phase 
displacement  between  the  impressed  voltage  and  current  are 
omitted  here  for  lack  of  space. 

The  diagrams  just  discussed  (Figs.  509  to  511)  apply 
equally  to  motors  having  inductive  compensation,  since  the 
local  drop  in  the  impedance  of  the  compensation  windings  is 
furnished  by  the  impressed  voltage  as  in  any  transformer. 

The  vector  diagram  of  an  inductively  excited  motor  such  as 
diagrammed  in  Fig.  501  is  practically  similar  to  that  shown  in 


Fig.  512.  — Vector  Diagram  of  a Repulsion  Motor  or  an  Inverted  Inductive  Series 

Motor. 


Fig.  509.  The  only  difference  in  the  diagrams  lies  in  the  fact 
that  the  secondary  induced  current  is  slightly  more  than  180°  be- 
hind the  primary  current  on  account  of  the  losses  in  the  winding 
circuits  ; hence  its  opposite  component  in  the  primary  current 
is  slightly  ahead  of  the  total  primary  current  ( OA , Fig.  509). 
As  the  field  induced  voltage  (OG' , Fig.  509)  is  at  an  angle  to  its 
current  equal  to  quadrature  plus  the  iron  loss  angle,  it  is  thrown 
slightly  to  the  left.  The  result  is  a slightly  lower  power  factor. 
With  A and  C of  Fig.  504  connected  to  the  line  and  the  brushes 
of  B short-circuited  there  results  a split  coil  repulsion  motor. 


886 


ALTERNATING  CURRENTS 


The  vector  diagram  for  such  a motor  may  be  drawn  quite  simi- 
larly to  that  of  Fig.  509,  since  the  speed-induced  counter- 
voltage is  transferred  directly  to  the  compensating  or  mutual 
inducing  coil  C.  Thus,  assume  that  current  OA  of  Fig.  512  is 
flowing  through  the  torque  and  mutual  coils  of  Fig.  496.  Cur- 
rent flows  in  the  armature,  lagging  slightly  less  than  180°  from 
OA , supposing  the  reluctance  of  the  path  of  the  flux  which 
links  the  mutual  coils  and  armature  coils  is  small.  The  voltage 
set  up  by  the  armature  rotation  is  in  phase  opposition  to  OM. 
This  induces  in  the  mutual  coil  0 a practically  equal  and  oppo- 
site voltage  OC.  The  self-inductive  voltage  of  the  field  GrF  is 
at  right  angles  to  OM.  Then  if  OF  is  the  total  local  drop  of 
voltage  due  to  magnetic  leakage  and  conductor  resistance,  the 
impressed  voltage  is  OJ , and  the  construction  is,  as  said,  sim- 
ilar to  that  of  Fig.  509. 

The  voltage  induced  in  the  field  coils  of  a series  motor  by  the 
torque  flux  is 


(i) 


where  n is  the  number  of  field  turns,  f the  frequency  of  the 
supply  current,  Xs  the  reactance  of  the  winding,  and  I the 
motor  current. 

If  the  magnetic  flux  were  constant  instead  of  alternating,  the 
counter-voltage  induced  in  the  armature  by  its  rotation,  with 
the  brushes  in  the  neutral  plane,  would  be 


F = a~ 
108 


in  which  a is  the  number  of  conductors  on  the  armature,  is 
the  total  flux  per  field  pole,  v is  the  number  of  revolutions  of 
the  armature  per  second,  p is  the  number  of  pairs  of  poles  in  the 
magnetic  field,  and  p'  is  the  number  of  paths  in  parallel  for  cur- 
rent to  flow  through  the  armature  winding.  But  the  flux  is  of 
sinusoidal  alternations  instead  of  being  constant,  and,  being 
the  maximum  flux  per  pole,  the  counter-voltage  set  up  in  the 
armature  by  its  rotation  is 


E = 


V2  x 108  P1 


(2) 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


887 


The  third  voltage  used  in  the  diagrams  represents  that  ab- 
sorbed by  resistance  and  leakage  reactance  and  is 

E,=  IZ*  (3) 

in  which  Zt  is  the  impedance  from  resistance  and  leakage  react- 
ance. 

If  Rc  is  a resistance  such  that  IRC  = Ec,  then 


7?  _ Ec  a<S>v  p_ 

' I V2  x 108  x I P1' 


(4) 


Putting  2 tv  for 

00 


a 

n 


and  S for 


v 

T 


gives  from  Equations  (1)  and 


The  motor  input  is 


the  output  is 
and  the  torque  is 


Rc  = -?-7  wSXf. 

irp' 

Pi  = El  cos  0, 
P0  = PRC, 

T RJ\ 

27 TV 


(5) 


208.  Commutation  and  Other  Characteristics  of  Commutating 
Alternating-current  Motors. — The  process  of  commutation  in 
alternating-current  motors  differs  from  that  of  direct-current 
machines  by  reason  of  the  voltages  induced  in  the  short-circuited 
coils  under  the  brushes,  caused  by  the  rapidly  alternating  char- 
acter of  the  field  flux  passing  through  them.  Thus,  to  prevent 
sparking,  it  is  necessary  to  neutralize  these  induced  voltages  to 
reasonable  values  or  reduce  the  resultant  currents  and  yet  per- 
form the  ordinary  function  of  reversing  the  current  in  the  coils 
while  short-circuited  under  the  brushes.  It  will  also  be  noted 
that  the  value  of  the  current  in  the  short-circuited  coils  is  de- 
pendent upon  two  conditions  at  each  instant  of  the  primary 
current  cycle.  Namely,  the  actual  currents  in  successively 
short-circuited  coils  depend  upon  the  load  and  the  instantaneous 
value  of  the  field  flux.  Thus,  at  the  instant  when  the  field 
flux  is  a maximum,  no  current  is  induced  in  the  short-circuited 
coils  due  to  its  change,  but  the  coils  enter  the  condition  of 
short  circuit  bearing  maximum  line  current ; and  when  the 
field  flux  is  zero,  a maximum  current  is  induced  in  the  short- 
circuited  coil,  but  the  line  current  is  zero. 


888 


ALTERNATING  CURRENTS 


The  short-circuited  coils  being,  with  respect  to  the  field  wind- 
ings, in  effect  transformer  secondaries  of  very  low  resistance, 
the  short  circuit  currents  become  so  great,  if  not  reduced  by 
some  special  construction,  as  to  seriously  demagnetize  the  fields 
and  interfere  with  the  operation  of  the  motor  by  destroying  its 
torque.  This,  taken  in  connection  with  the  liability  to  serious 
sparking,  makes  the  question  of  commutation  of  very  serious  im- 
portance in  designing  alternating-current  commutator  motors. 

One  of  the  methods  of  reducing  the  short-circuit  currents  is 
to  introduce  high  resistance  strips  into  the  leads  between  the 
commutator  bars  and  the  armature  windings.  This  is  useful  ; 
but  leads  of  sufficient  resistance,  having  large  enough  radiating 
surfaces  to  dissipate  the  heat  generated  in  them,  are  difficult  to 
insert  in  the  confined  space  available.  Therefore,  though  suit- 
able when  the  armature  is  moving  and  the  resistance  strips  are 
in  use  only  a small  part  of  a revolution,  they  are  liable  to  be 
destroyed  if  the  motor  is  stalled  or  fails  to  start  promptly. 
Nevertheless,  this  method  has  proved  valuable. 

Another  method  is  to  insert  reactance  coils  in  the  commutator 
leads  and  thus  avoid  the  heating  caused  by  the  previous  method, 
but  with  the  disadvantage  of  adding  to  the  self-inductance  of 
the  short  circuit  and  thus  making  commutation  difficult.  Vari- 
ous arrangements  have  been  proposed  of  this  nature.  One  of 
them  uses  a differentially  wound  coil  so  that  the  line  current 
largely  neutralizes  the  reactance  in  its  path  while  the  short- 
circuited  current  sets  up  full  reactive  flux  in  the  coil. 

Another  method  uses  a wide  space  between  commutator  bars 
and  a split  brush  between  the  halves  of  which  is  a reactance 
coil.  The  two  halves  of  the  brush  are  so  spaced  that  they 
touch  two  adjacent  bars,  and  hence  the  short-circuit  current 
must  flow  through  the  reactance.  Special  devices  may  be  ar- 
ranged so  that  the  line  current  is  not  affected  by  the  reactance. 

The  use  of  overcompensation  with  connections  of  the  motor 
windings  in  the  relations  of  a series  repulsion  motor  has  been 
found  desirable  for  reducing  the  transformer  voltage  in  the 
short-circuited  coils.  The  use  of  commutating  interpoles  placed 
over  the  brushes  has  also  been  found  advantageous.  These 
poles  can  have  windings  which  are  shunt-connected  across  the 
brushes  and  through  a variable  autotransformer  so  that  their 
strength  may  be  changed  with  the  speed  and  load.  A method 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


889 


shown  in  Fig.  513  of  connecting  the  commutating  poles  gives 
one  of  the  many  possible  arrangements.  Thus,  part  A of  the 
commutating  pole  winding  is  shunted  in  series  with  resistance 
B across  the  field  coil,  which  gives  a current  strength  in  A pro- 
portional to  the  field 
strength.  The  part  F 
of  the  same  winding 
is  connected  in  mag- 
netic opposition  to  A 
and  receives  current 
from  the  secondary  cir- 
cuit of  a small  trans- 
former 6f,  the  primary 
winding  of  which  is 
connected  across  the 
armature  brushes  and 
which  receives  current  in  proportion 
to  the  armature  speed.  Then  at  low 
speeds,  when  the  current  to  be  reversed 
in  the  armature  conductors  and  the  field 
flux  tending  to  induce  heavy  secondary 
currents  in  the  short-circuited  coils 
are  large,  the  commutating  poles  are 
strong.  When  the  speed  is  high  and 
the  armature  current  and  field  flux  are 
low,  the  commutating  pole  flux  is  low. 

The  reactance  Vindicated  in  series  with 
the  small  transformer  is  required  to  ad- 
just the  value  and  phase  of  the  current.* 

However,  since  commutating  poles 
or  overcompensation  are  effective  to  any 
extent  only  when  the  armature  is  rotating  and  hence  its  con- 
ductors are  cutting  the  commutating  flux,  such  schemes  require 
also  the  use  of  resistance,  or  an  equivalent  device,  in  the  leads 
between  armature  coils  and  commutator. 

The  commutation  of  repulsion  motors  is  better  when  the 
speeds  are  below  synchronism  than  when  the  speeds  exceed  syn- 
chronism, while  the  reverse  is  true  of  plain  series  motors  whether 
compensated  or  uncompensated. 

* McAllister,  Alternating  Current  Motors,  p.  302. 


Alternating-current  Motor 
having  Commutating  Poles 
connected  for  Automatic 
Strength  Variation. 


890 


ALTERNATING  CURRENTS 


The  operating  characteristics  of  a series  alternating-current 
motor  are  shown  in  Fig.  514.  As  in  a direct-current  series- 
motor,  the  torque  starts  at  a high  value  and  decreases  as  the 
speed  increases  and  the  current  and  field  flux  decrease.  The 
power  rises  rapidly  to  a maximum  when  the  product  of  torque 
and  speed  is  a maximum  and  then  decreases.  The  power  factor 
rises  rapidly  to  nearly  unity. 

From  the  formula  and  diagram  already  developed  (page  887) 

-X"  7T  ^ 

it  may  be  shown  that  tan  6 = = -IL  , approximately,  where 

nc  wSp 

6 is  the  angle  of  lag  in  a series-motor.  Then  to  reduce  0 to  a 


R.P.  M. 

Fia.  514.  — Operating  Characteristics  of  a Series  Alternating-current  Motor,  Compen- 
sated. 

minimum,  w,  S and  p should  be  as  great  as  possible  — a fact 
which  has  been  found  true  in  practice.  To  keep  the  power 
factor  at  the  highest  possible  value,  it  is  evidently  desirable  to 
reduce  the  air  gap  to  the  minimum  allowable  by  mechanical 
considerations. 

The  frequency  for  which  large  series-motors  are  built  in  this 
country  is  usually  25  cycles  per  second,  though  mam*  smaller  mo- 
tors of  the  series  and  repulsion  tj-pes  are  used  on  circuits  having 
a frequency  of  60  cycles  per  second.  Some  European  traction 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


891 


circuits,  on  which  commutating  alternating-current  motors  are 
used,  have  frequencies  of  15  periods  per  second  or  even  less. 

Alternating-current  motors  having  their  field  windings  con- 
nected in  shunt  with  the  armature  can  be  built,  though  diffi- 
culty is  experienced  in  keeping  the  field  flux  and  armature 
current  in  phase ; and  further  the  place  for  motors  of  this  type 
is  occupied  satisfactox-ily  by  the  single-phase  induction  motor 
or  repulsion  motor. 

The  efficiency  of  alternating-current  commutator  motors  can 
best  be  obtained  by  measuring  the  input  b}^  a wattmeter  and 
the  output  by  a rated  generator  or  other  mechanical  absorbing 
device.  However,  the  losses  can  .be  approximately  determined 
if  desired.  The  copper  losses  can  be  calculated.  The  field 
core  losses  cau  be  measured  by  a wattmeter  when  the  field  is 
excited,  the  armature  being  removed.  And  the  armature  core 
loss  and  other  rotating  losses  can  be  approximately  obtained  by 
driving  the  armature  by  a rated  motor  when  the  field  is  fully 
excited  for  the  load  and  speed  desired. 

209.  Frequency  Changers  and  Motor  Converters. — Frequency 
changers  are  much  used  in  this  country  for  transforming  low 
frequency  currents  to  higher  frequency  where  electric  power 
transmitted  from  a distance  is  needed  for  lighting. 

One  form  of  frequency  changer,  called  a Motor-generator, 
consists  of  a motor  operated  from  the  circuit  of  one  frequency 
and  driving  a generator  connected  with  the  circuit  of  the 
other  frequency.*  The  two  machines  are  usually  mounted 
on  one  bed  plate  and  have  a common  shaft,  thereby  making 
a motor-generator  set,  and  a starting  motor  is  sometimes  also 
mounted  on  one  end  of  the  shaft.  The  numbers  of  magnet 
poles  in  the  field  magnets  of  the  two  machines  must  be  in 
the  ratio  of  the  two  frequencies.  When  two  such  machines 
are  expected  to  operate  in  parallel  at  both  ends  of  the  sets,  it 
is  necessary  that  the  alignment  of  the  field  magnets  and  arma- 
ture coils  shall  be  exactly  corresponding  in  the  two,  or  exces- 
sive cross  currents  will  flow.  It  is  also  necessary  to  synchronize 
at  both  ends  of  the  set,  since  the  set  may  parallel  correctly  at 
one  end  and  yet  be  out  of  correct  phase  for  paralleling  at  the 
other  end.  In  this  case  it  is  necessary  to  slip  one  or  more  poles 
at  the  motor  end  by  opening  the  field  switch,  or  in  some  other 


* Art.  183. 


892 


ALTERNATING  CURRENTS 


manner  bring  the  machine  into  paralleling  phase  at  both  ends. 
The  number  of  points  in  one  revolution  of  the  shaft  of  such  a 
machine  at  which  paralleling  can  be  accomplished  at  both  ends 
of  the  set  is  equal  to  the  greatest  common  factor  of  the  numbers 
of  pairs  of  poles  respectively  in  the  field  magnets  of  the  motor 
and  the  generator.  The  generator  terminal  voltage  of  an  un- 
loaded set  which  is  to  be  put  in  parallel  with  a loaded  set  will 
also  not  be  in  exactly  the  same  phase  as  the  terminal  voltage  of 
the  loaded  set,  on  account  of  the  mechanical  lag  of  the  loaded 
motor  and  the  electrical  lag  of  the  loaded  generator,  even 
though  the  motors  have  been  brought  into  the  correct  relations 
for  paralleling  the  sets. 

A Frequency  converter  as  distinguished  from  a motor-gener- 
ator consists,  in  its  ordinary  form,  of  a synchronous  motor  driv- 
ing the  armature  of  an  induction  motor  backwards,  i.e.  against 
its  direction  of  rotation  as  a motor,  the  armature  winding  of 
the  synchronous  machine  and  the  field  winding  of  the  induc- 
tion machine  being  connected  to  the  supply  mains  of  lower 
frequency,  while  the  armature  winding  of  the  induction  ma- 
chine is  connected  to  the  higher  frequency  circuit.  The  con- 
nections are  shown  in  Fig.  515.  The  power  which  must  be  sup- 
plied by  the  synchronous  motor  in  driving  the  induction 
motor  armature  backward  is  equal  to  the  torque  exerted  be- 
tween the  induction  motor  field  flux  and  armature  current 
times  the  speed  of  the  armature,  while  that  transferred  by 
the  induction  motor  itself  is  that  which  would  be  trans- 
ferred if  the  armature  were  stationary  and  the  same  cur- 
rent flowed.  Therefore,  if  a is  the  frequency  of  the  supply 
mains  and  b is  the  frequency  of  the  secondary  mains,  the  in- 
duction motor  must  supply  j times  the  total  power  transformed 


and  the  synchronous  motor  must  supply 


times  the  total 


power.  Then,  if  it  is  required  to  convert  200  kilowatts  from 
25  to  60  cycles,  the  induction  motor  field  must  transform 
x 200,  or  83^  kilowatts,  while  the  synchronous  motor  must 
supply  116-|-  kilowatts  to  the  armature  shaft.  The  armature 
losses  must  be  supplied  in  equal  ratio.  The  voltage  given  out 

will  be  equal  to  - times  the  voltage  that  would  be  generated 
a 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


893 


in  the  armature  windings  if  they  were  stationary.  The  capacity 
of  synchronous  motor  and  induction  motor  field  winding  must 
be  then  in  the  proportion  given,  but  the  capacity  of  the  induc- 
tion motor  armature  must  be  that  of  the  entire  output.  The 
losses  of  the  armature  tend  to  be  high  on  account  of  the  high 
frequency  of  the  magnetic  cycles  to  which  its  core  is  subjected. 


SUPPLY  MAINS 


Fig.  515.  — Connections  of  a Frequency  Converter. 


A Motor-converter  is  a machine  for  converting  compara- 
tively high  voltage  alternating  currents  to  comparatively  low 
voltage  direct  currents  without  the  intervention  of  a transformer. 
It  consists  of  the  combination  of  an  induction  motor  and  a rotary 
converter  as  shown  in  Fig.  516.  The  armature  of  the  motor 
sends  polyphase  currents  directly  into  the  windings  of  the  con- 
verter armature,  but  slip  rings  are  not  needed,  as  the  machines 
are  connected  together  mechanically.  Suppose  that  the  two 


894 


ALTERNATING  CURRENTS 


machines  have  equal  numbers  of  poles,  then  when  the  induction 
motor  armature  reaches  one  half  synchronous  speed,  the  current 
it  sends  to  the  synchronous  converter  armature  has  a frequency 
equal  to  the  frequency  of  the  counter-voltage  induced  in  the 
converter  armature.  The  frequency  of  the  voltage  induced  in 


RESISTANCE  Fig.  516.  — Diagram  of  a Motor-converter. 


the  armature  conductors  of  the  induction  motor  falls  propor- 
tionally with  the  armature  speed,  from  the  frequency  of  the 
supply  circuit  to  zero,  as  the  armature  increases  in  speed  from 
standstill  to  synchronism.  The  voltage  likewise  falls  off  dur- 
ing this  change  of  speed.  At  the  same  time  the  frequency  of 
the  voltage  induced  in  the  armature  conductors  of  the  synchro- 

O J 


ASYNCHRONOUS  MOTORS  AND  GENERATORS 


895 


nous  converter  armature  rises  proportionally  with  the  armature 
speed,  from  zero  to  the  frequency  of  the  supply  circuit,  in  the 
same  range  of  speed,  and  the  voltage  itself  increases.  At  half 
synchronous  speed  the  induced  voltages  are  of  the  same  fre- 
quency. If  the  speed  tends  to  rise,  the  counter-voltage  of  the 
synchronous  converter  armature  tends  to  reverse  the  resultant 
voltage  in  the  circuit  of  the  armatures.  Therefore,  if  the  in- 
duction motor  and  synchronous  converter  are  of  the  proper 
relative  proportions  and  have  equal  numbers  of  field  poles,  the 
speed  will  be  maintained  equal  to  one  half  the  speed  corre- 
sponding for  the  induction  motor  to  synchronism  with  the 
supply  current.  Part  of  the  power  converted  by  the  synchro- 
nous converter  is  received  mechanically  through  its  shaft  from 
the  induction  motor  and  part  electrically  from  the  induction 
motor  armature  windings.  If  the  machines  have  unequal  num- 
bers of  field  poles,  the  speed  of  the  set  is  proportional  to  the 
ratio  of  the  number  of  poles  on  the  induction  motor  to  the  sum 
of  the  numbers  of  poles  on  the  two  machines. 

In  addition  to  permitting  the  use  of  high  alternating  voltages 
without  the  use  of  transformers,  the  motor-converter  has  the 
advantage  of  furnishing  a low  frequency  to  the  rotary  portion 
of  the  apparatus. 

210.  The  Mercury  Vapor  Rectifier.  — The  mercury  vapor 
rectifier  or  converter  has  come  much  into  use  for  operating 
series  direct-current  arc  lamp  circuits,  and  charging  storage 
battery  circuits.  It  consists  of  an  exhausted  globe  as  shown 
in  Fig.  517.  A small  amount  of  mercury  is  placed  in  the  tube 
or  globe,  and  when  the  latter  is  shaken,  causing  the  mercury  to 
bridge  across  the  terminals  A and  B,  a current  flows  from  the 
alternating-current  mains  between  the  terminals.  This  frees 
mercury  vapor,  and  current  flows  alternately  into  the  terminals 
O and  O'  and  out  of  the  terminal  A.  By  virtue  of  the  peculiar 
property  of  the  vapor  in  connection  with  the  terminal  electrode, 
current  apparently  cannot  flow  out  of  the  terminals  O and  O', 
so  that  it  flows  in  on  alternate  half  cycles.  Consider  an  instant 
when  it  is  flowing  into  0:  It  then  passes  from  the  iron  or 

carbon  electrode,  through  the  vapor,  into  the  mercury  at  the 
bottom,  out  of  terminal  A , through  the  storage  battery  E, 
through  the  reactance  coil  F,  and  thence  to  the  alternating- 
current  return  main.  Upon  the  next  half  cycle,  the  current 


896 


ALTERNATING  CURRENTS 


flows  into  the  rectifier  at  C' , out  at  A,  and  thence  through  the 
battery  in  the  same  direction  as  before,  then  through  F'  and  out 
to  the  other  main. 

A drop  of  voltage  of  about  15  volts  occurs  when  current  passes 

through  the  tube,  which 
makes  it  quite  efficient 
for  supplying  series  arc 
lighting  circuits,  in  which 
the  current  is  low  and  the 
voltage  high.  For  such 
circuits  the  alternating 
current  is  usually  sup- 
plied to  the  rectifier  by  a 
constant-current  trans- 
former.* Polyphase 
tubes  are  also  used  for 
charging  storage  batteries 
and  similar  purposes.  In 
a tri-phase  tube  three 
positive  electrodes  similar 
to  C and  C are  used. 
The  secondary  windings 
of  the  supply  trans- 
formers are  connected  in  wye.  The  free  battery  terminal  is 
connected  to  the  neutral  point  and  the  three  electrodes  are 
connected  to  the  three  wye  terminals. 

Electrolytic  rectifiers  are  also  used  to  some  extent.  They 
are  based  on  the  fact  that  an  aluminum  electrode  in  a cell  will 
pass  a current  only  in  one  direction  (i.e.  that  which  makes  the 
aluminum  the  anode)  unless  the  voltage  exceeds  a certain  criti- 
cal figure.  This  peculiarity  of  aluminum  is  also  made  use  of  in 
the  electrolytic  lightning  arresters,  now  largely  used  on  high 
voltage  transmission  circuits.  For  this  purpose  what  are  equiv- 
alent to  long  strings  of  aluminum  cells  are  joined  in  series  from 
a line  wire  to  ground  or  from  one  line  wire  to  another.  When  a 
current  is  passed  through  them,  they  quickly  take  a property  of 
very  high  resistance.  This  property  gradually  disappears  with 
time,  when  the  arrester  must  be  “ recharged,”  by  having  a suit- 
able voltage  impressed  upon  it  for  a short  time. 

* Art.  142. 


Fig.  517.  — Single-phase  Mercury  Vapor  Rectifier. 


CHAPTER  XIII 


SELF-INDUCTANCE,  MUTUAL  INDUCTANCE,  AND  ELECTRO- 
STATIC CAPACITY  OF  PARALLEL  WIRES.  SKIN  EFFECT 

211.  The  Self-inductance  of  Parallel  Wires.  — The  self-in- 
ductance of  two  parallel  wires  hanging  upon  a pole  line,  con- 
tained in  a cable,  or  otherwise,  has  much  influence  on  the 
processes  of  long  distance  power  transmission,  long  distance 
telephony,  and  long  distance  telegraphy.  In  the  ordinary 
alternating-current  systems  for  electric  lighting  and  for  trans- 
mission of  power  over  relatively  short  distances,  the  effects  of 
the  self-inductance  of  the  transmission  lines  are  not  particularly 
serious,  but  even  in  those  instances  it  is  important  to  be  able  to 
predetermine  the  nature  and  magnitude  of  the  effects. 

An  expression  for  the  self-inductance  of  two  parallel  wires 
may  be  developed  thus  : 
conductors  A and  A', 

Fig.  518,  form  a circuit 
of  indefinitely  great 
length.  Let  I be  the 
amperes  of  current  flow- 
ing through  the  con- 
ductors, r the  radius  o.f 
each  conductor  meas- 
ured in  centimeters,  and 
d the  distance  between 
the  axes  of  the  con- 
ductors also  measured 
in  centimeters,  and  as- 
sume the  current  to  be 
uniformly  distributed 
over  the  cross  section  of 
each  of  the  conductors. 

Also  let  fi  and  ///  be  respectively  the  magnetic  permeability 
of  the  medium  surrounding  the  conductors  and  of  the  material" 
composing  the  conductors.  The  intensity  of  the  magnetic  field 
3 m 897 


Suppose  that  two  parallel  cylindrical 


/ / 


<— 1 crriT> 

PLANE 

i 

d 

PLANE 

/ 

/ 

Fig.  518. — Diagram  for  studying  Self-inductance 
of  Parallel  Wires,  consisting  of  Two  Conductors 
A,  A'  of  Indefinite  Length  cut  at  Right  Angles  by 
Two  Imaginary  Planes. 


898 


ALTERNATING  CURRENTS 


(Ha)  measured  in  dynes  at  a point  outside  of  the  conductor  A, 
at  a distance  a centimeters  from  its  center  and  due  to  the 
current  in  A , is 


H = 


21 
10  a 


(1) 


This  may  be  readily  understood  from  the  fact  that  the  lines 
of  force  due  to  the  current  in  the  straight  conductor  are  circles 
with  their  planes  perpendicular  to  the  conductor,  and  if  the 
magnetic  force  at  any  point  in  space  at  a distance  a centimeters 
from  the  conductor  is  Fa  dynes,  the  work  done  against  this 
force  in  moving  a unit  magnet  pole  around  the  conductor  is 
W = 2 7 raFa  ergs.  In  this  case  the  electromagnetic  intensity, 
and  therefore  the  force  exerted  on  a unit  pole,  is  perpendicular 
to  the  conductor,  since  the  lines  of  force  are  circles  with  their 
planes  perpendicular  to  the  conductor.  Also,  the  work  done 
in  carrying  a unit  pole  around  the  conductor  is  equal  to 

W = 4:7m—  in  any  case  in  which  n is  the  number  of  times 

10  J 


the  current  is  circled  and  the  current  is  measured  in  amperes. 
The  symbol  n stands  for  the  number  of  turns  in  the  coil  if  the 
conductor  concerned  were  a coil ; but  in  this  case  n = 1,  and 
therefore 


2 iraFa 


4 7 rl 
10  ’ 


or 


F=H  = 


21 

10  a' 


The  magnetic  density  (i?a)  in  lines  of  force  per  square  centi- 
meter at  the  point  a is  therefore 

& <2> 

Now,  considering  a space  cut  out  by  two  planes  perpendicular 
to  the  axes  of  the  conductors  and  one  centimeter  apart  (see 
Fig.  518),  within  this  space  at  a distance  a centimeters  from 
conductor  A a number  of  lines  of  force  equal  numerically  to 

Bada  = d^a=m^ 

10  a 


‘will  pass  through  a radial  width  da  centimeters  ; and  the  total 
number  of  lines  of  force  set  up  by  the  current  in  A which  pass 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


899 


through  the  area  bounded  on  two  sides  by  the  surface  of  A and 
the  center  of  A'  and  on  the  other  two  sides  by  the  planes,  is 


rd2  filda 
Ar  10  a 


(3) 


At  any  point  p within  conductor  A and  at  a distance  of  b 
centimeters  from  the  center  of  A,  the  magnetic  intensity  will 
be  the  same  as  though  the  current  within  a circle  of  radius  b 
centimeters  were  condensed  at  the  center  of  the- conductor,  as 
far  as  the  magnetic  effect  of  the  current  in  conductor  A is  con- 
cerned. Since  the  magnetic  effect  at  the  point  p of  the  uniform 
layer  of  the  current  in  the  conductor  outside  of  radius  b centi- 
meters will  be  zero,  the  strength  of  field  at  p may  be  written 


2 I irb 2 _ 2 bl 

105  X 7T?  ~ ToV2' 


(4) 


and  the  magnetic  density  at  p is 

(5) 

Hence  within  the  conductor  and  between  the  two  planes  one 
centimeter  apart,  a number  of  lines  of  force  numerically  equal  to 


Bh  = 


2 n’bl 
10  r2  ' 


Bbdb  = dA>b  - - 


2 p'bldb 
10  r2 


will  pass  through  a radial  width  db.  But  this  flux  does  not 
link  with  the  whole  current  in  the  conductor  A , but  only  with 
7rb2 

the  current  equal  to  — - 1 amperes  which  is  within  the  circle 

having  a radius  of  b centimeters.  The  product  of  the  current 
with  the  number  of  lines  of  force  inclosed  by  it,  giving  the 
magnetic  linkages  for  current  within  the  whole  conductor, 
therefore  is 

2 p'lbdb  1/J  r6. 

«/o  7 rr2  10  r2  2 10 

The  self-inductance  of  conductor  A given  in  henrys  for  a 
centimeter  of  length  is  equal  to  10-8  times  the  number  of  lines 
of  force  linking  the  current  when  it  has  a value  of  one  ampere ; 
and,  summing  the  effects  of  the  magnetic  field  without  and 


900 


ALTERNATING  CURRENTS 


within  the  conductor  by  adding  (3)  to  (6)  when  I is  one  am- 
pere, and  multiplying  by  10~8  gives 

i=(2/*log^4-|/*,)l0-»,  (7) 

which  is  the  self-inductance  in  henrys  per  centimeter  of  con- 
ductor length. 

The  effect  of  the  return  conductor  A!  in  a single-phase  cir- 
cuit is  to  exactly  double  the  flux  which  is  linked  or  inclosed  by 
the  current,  and  therefore  the  self-inductance  per  centimeter  of 
length  of  circuit  is  the  same  as  the  self-inductance  for  two  cen- 
timeters length  of  conductor. 

Given  in  terms  of  common  logarithms  the  self-inductance  per 
centimeter  of  length  of  conductor  is 

L=  ^4.605  ti  log-  + O'9 ; 

and  in  terms  of  1000  feet  of  conductor  the  self-inductance  is 
Lx  = ^.1404  n log  - + .01524  fx'^j  10~3  henrys 

or  .1404  /a  log  - + .01524  /x ' millihenry. 

r 

When  the  conductors  are  of  copper  suspended  in  the  air  or 
embedded  in  the  usual  insulating  materials,  fx  — /x'  = 1 and  the 
self-inductance  per  1000  feet  of  conductor  is 

X = .1404  log-  -f-  .01524  millihenry. 
r 

As  a rule  the  value  of  2 log  - derived  from  the  distances 

r. 

apart  of  diameters  of  wires  in  ordinary  overhead  electric  cir- 
cuits is  quite  large  compared  with  the  term  4,  but  in  the  case 

of  conductors  in  underground  cables  the  ratio  - may  be  so 

r 

small  that  the  constant  term  ^ comprises  a large  part  of  the 
self-inductance.  In  underground  cables,  however,  the  con- 
ductors are  so  close  together  that  the  uniformity  of  distribu- 
tion of  the  currents  over  the  cross  section  of  each  conductor 
cannot  be  relied  upon,  as  the  conductors  will  influence  each 
other  and  the  results  may  be  modified. 

The  following  table  gives  the  resistances  and  self-inductances 
of  wires  of  different  sizes  at  a number  of  distances  apart. 


Table  showing  Self-inductances  of  Solid  Cylindrical  Copper  Conductors  at  Various  Distances  Apart, 

Measured  from  Center  to  Center 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


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902 


ALTERNATING  CURRENTS 


In  the  foregoing  discussion  of  the  self-inductance  of  two  paral- 
lel conductors  the  upper  limit  of  integration  in  Equation  (3)  is 
taken  as  3,  which  carries  the  integration  from  the  edge  of  con- 
ductor A to  the  center  of  conductor  A'  for  the  magnetic  link- 
ages resulting  from  flux  outside  of  the  conductor  A.  This 
is  obviously  correct  for  each  thin  cylindrical  layer  of  current 
in  conductor  A',  and  is  therefore  correct  for  the  whole. 

In  case  the  solid  cylindrical  conductors  are  of  different  diam- 
eters, the  formula  for  the  self-inductance  per  centimeter  of 
conductor  length  becomes 

L = (/*  loge  — -f  ^y^llg-9,  (8) 

V rir2  Z J 

in  which  rx  and  r2  are  the  radii  of  the  two  conductors. 

The  equation  for  parallel  hollow  cylindrical  conductors  is, 
per  centimeter  of  conductor  length, 


L = 


i fl2  1 f r,2  — 3(V,)2  4(/,)4  i r,  1 

/*  loffe — + t Mi  + rA~n~^T2 lo^ 

rxr2  4 l r{  — {r\y  [rp  — (»i)yr  r\  J 

10-9,  (9) 


. 1 j rJ 


3(V2)2  | 


(^2)2  [^2-(^2)2]2 


l0ge^} 


2 1 -> 


in  which  /x,  fiv  /u,2  represent  the  magnetic  permeabilities  respec- 
tively of  the  medium  surrounding  the  conductors  and  the  two 
conductors,  rv  r2  represent  the  external  radii  of  the  two  con- 
ductors, r'v  r\  represent  the  internal  radii  of  the  hollow 
cylinders.* 

The  foregoing  formulas  relate  only  to  cjdindrical  wires,  but 
apply  with  an  ample  accuracy  to  conductors  of  other  forms  of 
cross  section,  such  as  square  or  oval,  provided  the  distance 
between  the  conductors  is  large  compared  with  their  thickness 
through.  In  the  case  of  flattened  conductors  in  underground 
cables  where  the  conductors  are  near  together,  these  formulas 
can  he  applied  approximately  in  lien  of  satisfactory  exact 
formulas  which  have  not  been  developed  on  account  of  the 
complexity  introduced  in  the  integrations  when  the  conductors 
deviate  far  from  cylindrical  form. 

If  the  currents  are  not  uniformly  distributed  over  the  cross 
section  of  the  conductors,  as  may  be  the  case  if  conductors  A 

* See  Maxwell,  Electricity  and  Magnetism,  Art.  685. 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY  903 

and  A!  lie  close  together  and  are  so  thick  that  the  magnetic 
flux  from  each  one  sets  up  an  appreciable  redistribution  of  the 
flux  in  the  metal  of  the  other,  the  inductive  reactance  com- 
puted from  the  formulas  may  go  quite  wide  of  the  truth  if  the 
reactance  is  taken  as  2 7 rfL  and  L computed  from  the  formulas. 

In  case  the  alternating  currents  are  of  rather  high  fre- 
quency and  the  diameters  of  the  conductors  are  considerable, 
the  currents  tend  to  forsake  the  central  part  of  each  conductor 
and  become  of  greater  density  toward  its  surface  on  account 
of  the  so-called  Skin  effect,*  and  an  approximation  of  the 
self-inductance  may  then  be  made  by  means  of  formula  (9). 
When  the  frequency  is  sufficiently  great  to  cause  the  currents 
to  concentrate  substantially  in  a thin  cylindrical  film  of  the 
metal  at  the  surface,  the  inner  and  outer  radii  of  the  cylindrical 
conductor  become  equal  to  each  other. 

Formula  (7)  may  be  written  in  the  following  form  in  case 
the  conductors  are  of  the  same  radius  and  the  same  material 
and  are  sufficiently  far  apart  compared  with  their  thickness 
so  that  they  do  not  affect  the  current  distribution  in  each  other, 

L = C 2/xloge?  + %')10-9, 

r 2 

in  which  v is  a variable  coefficient  with  limits  from  unity  when 
the  currents  are  uniformly  distributed  over  the  cross  section 
to  zero  when  the  frequency  is  high  enough  to  bring  the 
currents  to  the  surface. 

The  minimum  value  for  the  self-inductance  of  a circuit 
composed  of  two  indefinitely  long  cylindrical  conductors 
obviously  occurs  when  the  conductors  are  separated  by  the  least 
practicable  distance.  If  the  current  can  be  assumed  to  be 
uniformly  distributed  over  their  cross  sections,  Equation  (7) 
reduces  for  the  minimum  value  to 

L = (2  ix,  loge  2 + 1 n’)  10-9  = (1.386  n + .5  /*')  10~9, 

and  when  and  \jJ  are  equal  to  unity  this  becomes  Z=  1.886 
x 10"9  henrys  per  centimeter  of  conductor  length.  This 
reduces  to  L = 0 in  the  limiting  case  of  the  two  conductors 
very  close  together  and  carrying  currents  of  such  high  fre- 
quency that  the  currents  concentrate  wholly  on  the  surface, 


* Art.  212. 


904 


ALTERNATING  CURRENTS 


since  in  that  case  (the  conductors  being  so  close  together)  the 
currents  in  the  conductors  react  on  each  other  so  that  the  cur- 
rents in  the  tube  conductors  come  to  occupy  positions  in  narrow 
strips  of  the  surfaces  facing  each  other. 

When  the  conductors  of  equal  cross  section  are  concentric 
with  respect  to  each  other  and  the  inner  conductor  is  solid,  a 
similar  integration  shows  that  the  self-inductance  of  the  pair 
of  conductors  per  centimeter  of  length  of  the  cable  is 


2 log. 


R ( R2  + r2 \2 

r V r2  / 


log. 


where  r is  the  radius  of  the  inner  conductor  and  R is  the 
radius  of  the  inner  cylindrical  surface  of  the  external  con- 
ductor. 

In  case  the  inner  conductor  is  also  tubular,  the  distribution 
of  the  current  is  limited  to  the  area  of  a hollow  cylinder  and 
the  self-inductance  of  the  concentric  cable  per  centimeter  of 
length  of  the  cable  then  becomes 


L = 


2 log, 


_i 

r ( r 2 — rp2)2 


R 


log,  - + Ri  log  — - - (rx 2 + i^Xr2-^2)  [ 

ri  /li  > . 


10~9, 


in  which  r1  and  r are  respectively  the  inner  and  outer  radii  of 
the  inner  conductor  and  R1  and  R are  the  inner  and  outer  radii 
of  the  outer  conductor. 

Prob.  1.  A pair  of  solid  electric  light  wires  are  erected  on  a 
pole  line  15  inches  apart  center  to  center  for  a distance  of  12 
miles.  The  wires  are  of  0000  B.  & S.  gauge.  What  is  the 
self-inductance  of  the  loop  composed  of  these  two  conductors  ? 
What  is  the  inductive  reactance  of  the  loop  at  a frequency  of 
25  cycles  per  second  ? 

Prob.  2.  A concentric  underground  cable  3000  feet  long  is 
composed  of  a solid  No.  6 B.  & S.  gauge  wire  surrounded  by  an 
external  conductor  composed  of  a spiral  wrapping  of  fine  wire 
but  giving  the  effect  of  a hollow  cylinder  of  300  mils  internal 
radius  with  the  same  conducting  cross  section  as  the  inner  wire. 
The  insulation  between  these  two  conductors  is  composed  of 
* Russell,  Alternating  Currents,  Vol.  1,  pp.  53-54. 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


905 


oiled  paper  of  which  the  permeability  majr  be  considered  unity. 
What  is  the  self -inductance  of  the  loop  composed  of  these  two 
conductors? 

It  must  be  remembered  that  the  foregoing  formulas  for  the 
self-inductance  of  parallel  wires  are  applicable  only  to  lengths 
which  are  parallel  for  such  large  distances  compared  with  the 
distances  apart  of  the  conductors  that  the  ends  of  the  loops 
need  not  be  considered,  and  tliey  are  not  applicable  to  coils  or 
to  wires  with  frequent  turns  in  them,  inasmuch  as  the  magnetic 

2 I 

force  at  the  turns  does  not  respond  to  the  condition  Ha  — 

10  a 

The  equations  are  sufficiently  accurate  in  their  representation 
of  the  facts  to  meet  the  requirements  of  most  computations 
with  respect  to  overhead  and  underground  electric  light  and 
power  circuits,  telephone  circuits  and  telegraph  circuits  ; but 
may  prove  seriously  in  error  if  applied  to  wiring  on  switch- 
boards or  inside  of  buildings  on  account  of  tbe  many  turns 
which  may  exist  in  such  wiring,  and  also  may  prove  seriously 
in  error  if  applied  to  circuits  composed  of  conductors  deviating 
far  from  cylindrical  form.  Flattening  the  conductors  tends  to 
decrease  the  self-inductance.  Also  dividing  the  total  current 
between  two  circuits,  each  having  cylindrical  conductors  of 
half  the  total  cross  section,  reduces  the  reactive  voltage  in  the 
circuit  because  the  current  per  circuit  is  reduced  to  a greater 
degree  than  the  self-inductance  is  increased. 

Overhead  wires  of  considerable  thickness  and  the  conductors 
of  underground  cables  of  greater  cross  section  than  No.  6 B. 
& S.  gauge  wire  are  usually  made  by  stranding  a sufficient  num- 
ber of  smaller  wires  to  afford  flexibility  while  maintaining  the 
desired  aggregate  cross  section.  In  applying  the  formulas  to 
such  conductors  it  is  usual  to  assume  that  they  give  the  same 
magnetic  effects  as  smooth,  cylindrical  conductors  of  equal 
cross  section.  The  error  introduced  by  this  assumption  is 
difficult  to  determine  for  any  particular  instance,  as  it  depends 
upon  the  various  proportions  entering  into  the  strand.  It  is 
usually  in  the  direction  of  a positive  error,  that  is,  it  makes  the 
computed  self-inductance  somewhat  too  large,  but  the  error  is 
usually  quite  small. 

The  foregoing  formulas  all  deal  with  circuits  composed  of 
parallel  conductors  completing  a loop.  When  a conductor  is 


906 


ALTERNATING  CURRENTS 


vertical  with  respect  to  the  earth  and  removed  from  parallel 
conducting  bodies,  a flow  of  current  may  occur  through  the 
effects  of  radiation  or  of  electrostatic  capacity.  In  this  case 
the  self-inductance  per  centimeter  of  length  of  a cylindrical 
conductor  is  of  the  form  of 


in  which  R is  the  geometric  mean  distance  from  each  other  of 
the  elements  of  current  in  the  conductor  and  is  equal  to  re-*  or 
r ( r being  the  radius  of  the  wire),  depending  upon  whether  the 
current  is  uniformly  distributed  over  the  cross  section  of  the 
conductor  or  is  concentrated  on  its  surface. 

When  the  loop  composed  of  two  conductors  carries  currents 
which  are  in  different  phases  in  the  conductors,  as  is  the  case 
with  a loop  composed  of  either  pair  of  wires  of  a three-phase 
circuit,  the  self-inductive  voltage  introduced  into  the  loop  is 
not  equal  to  the  reactance  of  the  loop  multiplied  by  the  current 
in  one  of  the  wires.  The  currents  in  the  two  conductors  being 
not  in  the  same  phase,  the  maximum  magnetic  flux  set  up  in 
the  loop  is  proportional  to  their  vector  sum  instead  of  their 
algebraic  sum.  The  self-inductive  voltage  introduced  in  the 
loop  in  such  cases  is  therefore  equal  for  a balanced  circuit  to 
the  loop  reactance  per  unit  length  of  conductor  (as  given  above) 


n is  the  number  of  phases.  This  voltage  is  in  quadrature  with 
the  vector  representing  the  sum  of  the  two  currents.  In  the 
case  of  a symmetrically  placed  three-phase  circuit,  a convenient 
manner  of  computing  the  effect  of  self-induced  voltage  on  the 
regulation  of  the  circuit  is  to  consider  the  voltage  per  conductor, 
which  is  equal  to  the  conductor  reactance  multiplied  by  the 
conductor  current  and  acts  in  quadrature  to  the  conductor  cur- 
rent. This  voltage  is  opposite  to  the  reactive  drop  along  the 
wire  which  is  measured  between  the  conductor  and  the  neutral 
point. 

Prob.  1.  The  parallel  conductors  of  a balanced  three-phase 
circuit  are  each  composed  of  a No.  0 B.  & S.  gauge  wire  and  are 
located  at  the  corners  of  an  equilateral  triangle  so  that  they  are 

* Maxwell,  Electricity  and  Magnetism , Arts.  685-693.  Steinmetz,  Transient 
Electric  Phenomena  and  Oscillations , Chap.  VIII,  Sec.  III. 


current  per  conductor  and 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


907 


36  inches  apart,  center  to  center.  The  length  of  each  conductor 
is  10  miles,  and  each  carries  a current  of  50  amperes.  What 
is  the  value  of  the  self-inductive  voltage  induced  per  con- 
ductor ? What  is  the  value  of  the  self-inductive  voltage  in- 
duced in  a loop  composed  of  two  of  the  conductors  ? 

Prob.  2.  In  Prob.  1,  suppose  the  voltage  between  conductors 
at  the  receiving  end  of  the  line  is  10,000  volts,  what  is  the  delta 
voltage  and  also  what  is  the  wye  voltage  at  the  generator, 
assuming  that  the  onl}r  causes  contributing  to  line  drop  are 
resistance  and  self-inductance  ? 

212.  The  Distribution  of  Current  in  a Wire.  — The  distribu- 
tion of  current  over  the  normal  cross  section  of  a homogeneous 
conductor  along  which  it  flows  is  uniform,  provided  the  current 
is  steady,  as  this  is  the  distribution  that  requires  the  smallest 
voltage  to  cause  a given  current  to  flow.  The  proof  of  this 
theorem  is  as  follows:  The  total  power  lost  in  heating  the  con- 
ductor is,  according  to  Joule’s  law,  I2R,  where  R is  equal  to 

Z,  p,  and  A being  respectively  the  length  in  centimeters,  the 

specific  resistance  in  ohms  per  centimeter  cube,  and  the  area  in 
square  centimeters  of  the  conductor.  Considering  the  con- 
ductor to  be  divided  into  elementary  filaments  of  equal  area 
and  of  resistance  r,  the  current  flowing  in  one  of  these  is  i 
and  the  power  lost  in  it  is  i2r.  The  total  power  lost  in  the 
conductor  is  2i2r,  and  the  conditions  afford  the  two  simulta- 
neous equations 

2Z2r  = im, 

= I. 


These  conditions  can  be  simultaneously  fulfilled  only  when  the 
currents  in  the  filaments  are  all  equal  to  each  other,  and  the 
distribution  of  the  current  over  the  cross  section  of  the  con- 
ductor is  therefore  uniform.  If  the  filamentary  currents  were 
not  uniform,  the  values  of  ir  for  the  different  filaments  would 
differ  from  each  other  and  from  IR,  which  would  cause  a diver- 
sion of  current  from  one  filament  to  another  until  the  uniform 
distribution  was  again  reached. 

The  theorem  of  uniform  distribution  over  the  cross  section 
applies  to  steady  currents  as  just  shown,  hut  it  does  not  apply 
to  alternating  currents,  as  the  effect  of  mutual  induction  be- 
tween filaments  comes  in  to  disturb  the  distribution.  Suppose 


908 


ALTERNATING  CURRENTS 


a homogeneous  cylindrical  conductor  divided  into  concentric 
cylindrical  elements  of  equal  cross  sections,  then  by  the  formulas 
developed  in  the  preceding  article, 


_2  ill- 


<j>  _ log  2 + 

10  S r 10 


I 


which  exhibits  the  flux  within  a radius  d which  is  set  up  by  a 
steady  current  I flowing  in  the  conductor,  and  shows  that  a 
greater  number  of  lines  of  force  surround  the  central  element  of 
the  conductor  than  surround  the  elements  nearer  the  surface, 
when  the  current  is  uniformly  distributed  over  the  cross  section 
of  the  conductor.  In  fact,  when  the  current  is  uniformly  distrib- 
uted over  the  cross  section  of  the  conductor,  the  element  com- 
posing the  outside  of  the  cylindrical  conductor  is  surrounded 


by  ^ less  lines  of  force  than  the  central  element. 


When  the 


current  flowing  through  the  conductor  is  alternating,  however, 
the  uniformity  of  distribution  is  disturbed  because  a counter- 
voltage is  set  up  in  each  element  which  is  equal  to  — where 

dt 

<j>f  is  the  flux  which  is  set  up  by  the  current  and  surrounds  the 
filament  under  consideration. 

Since  (f)f  increases  from  the  outside  of  the  wire  towards  the 
central  elements,  unless  the  current  is  all  concentrated  on  the 
outer  surface  of  the  conductor,  the  counter-voltage  is  greatest 
at  "the  center  and  least  at  the  surface  of  the  conductor.  Con- 
sequently there  is  a tendency  for  the  current  to  forsake  the 
center  of  the  conductor  and  to  seek  a place  nearer  the  surface. 
This  tendency  is  directly  proportional  to  the  frequency  of  the 
cycles  of  flux  when  the  current  is  sinusoidal.  This  tendency  is 
opposed  by  the  tendency  of  the  current  to  a distribution  which 
will  give  the  least  loss  of  energy,  and  alternating  current  there- 
fore distributes  itself  in  a conductor  so  that  these  two  tenden- 
cies are  balanced  against  each  other,  for  which  reason  the  current 
becomes  distributed  over  the  cross  section  so  that  its  density 
increases  from  the  center  to  the  surface  of  the  conductor. 
This  makes  an  increase  in  the  actual  resistance  to  the  flow  of  al- 
ternating current , and  in  the  loss  of  energy  caused  by  the  current 
flowing  through  the  conductor. 

The  resistance  i?a,  which  a straight  conductor  of  length  l ex- 
poses to  an  alternating  current,  in  ratio  to  the  true  resistance 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


909 


(or  resistance  which  the  conductor  exposes  to  a steady  current) 
R,  may  be  calculated  by  the  following  formula  in  which  Ra 
and  R are  given  in  ohms,  f in  periods  per  second,  and  l in 
centimeters,* 


Ra_X  , 

R 12  U0  SR) 


2 tt/AV 

. 10  v ) 


JLr^fL Y+  etc 

lSOUO9^  2440U09i27 

/X4  f 2 tt/AV  , p?  (2  7rfA\6  , 

180V  109p  7 + 2440V  109p  7 6 


In  the  last  expression  A is  the  area  in  square  centimeters  of 
the  cross  section  of  the  conductor  and  p is  the  specific  resist- 
ance of  the  material  measured  in  ohms  per  centimeter  of  length 
and  square  centimeter  of  cross  section  (per  centimeter  cube). 

When  the  material  of  the  conductor  is  copper,  fi  = l,  and  the 
formula  may  be  written  approximately  for  copper  conductors, 
in  which  p = 16  x 10~7  ohms  for  the  centimeter  cube, 

^ = 1 + 30(dyio-9)2, 

R 


where  cP  is  the  number  of  circular  mils  in  the  cross  section. 
For  other  conductors  the  approximate  formula  is 

•|a  = l + 30^^10-9J, 

in  which  p is  the  specific  resistance  of  copper,  p'  the  specific 
resistance  of  the  material  composing  the  conductor  concerned, 
and  p is  the  permeability  of  the  conductor.  When  d2fp  in 
these  approximate  formulas  exceeds  5 x 109,  the  resistance 
which  a cylindrical  conductor  offers  to  alternating  currents  is 
approximately  equal  to  the  true  resistance  of  a tube  of  the  same 
material  with  its  outside  diameter  equal  to  that  of  the  conductor 
and  with  the  thickness  of  its  wall  equal  to  2500/V/mils. 

The  following  table  gives  the  ratio  of  Ra  to  R for  various 
values  of  the  product  d2f  for  cylindrical  conductors,  d being 
the  diameter  in  mils. 


* Maxwell,  Electricity  and  Magnetism , Vol.  II,  Chap.  13  ; Rayleigh , On  Self- 
Induction  and  Resistance  of  Straight  Conductors,  Philosophical  Magazine , 
May,  1886,  p.  381. 


910 


ALTERNATING  CURRENTS 


Product  of  Circular 

Ha 

Product  of  Circular 

Mils  and  Frequency 

It 

Mils  and  Frequency 

R 

10,000,000 

1.003 

70,000,000 

1.15 

15,000,000 

1.007 

80,000,000 

1.19 

20,000,000 

1.012 

90,000,000 

1.24 

30,000,000 

1.03 

100,000,000 

1.30 

40,000,000 

1.05 

125,000,000 

1.47 

50,000,000 

1.08 

150,000,000 

1.67 

60,000,000 

1.11 

Infinite 

GO 

This  table  shows  that  with  an  increase  of  the  frequency  of 
the  circuit  or  of  the  diameter  of  the  conductor,  or  of  both,  the 
energy  loss  per  ampere  flowing  in  the  conductor  increases,  and 
that  the  energy  loss  may  become  very  great  (as  though  the 
current  wholly  forsook  the  middle  parts  of  the  conductor  and 
confined  itself  to  a thin  exterior  layer)  when  the  product  of  the 
frequency  of  the  current  and  the  square  of  the  conductor 
diameter  becomes  very  great.  On  account  of  this  apparent 
concentration  of  the  current  in  a superficial  skin  of  the  wire  the 
phenomenon  is  commonly  called  Skin  effect. 

The  current  density  at  any  point  within  a conductor  is  shown 
by  Thomson  to  fall  off  in  going  from  the  surface  inwards,  in 
the  ratio  of 

where  x is  the  distance  of  the  point  from  the  surface.*  Gray  f 
points  out  that,  however  great  the  diameter  of  a wire  may  be, 
its  resistance  to  alternating  currents  of  different  frequencies 
will  never  be  less  than  the  true  resistance  of  a wire  of  the 
diameter  in  centimeters  given  in  the  following  table : 


Frequency 

Copper 

Lead 

Iron,  h = 300 

80 

1.43 

4.98 

.195 

120 

1.17 

4.08 

.159 

160 

1.02 

3.52 

.138 

200 

.91 

3.16 

.123 

* J.  J.  Thomson,  Recent  Researches  in  Electricity  and  Magnetism,  pp.  260 
and  281. 

t Absolute  Measurements  in  Electricity  and  Magnetism,  Yol.  II,  p.  338. 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


911 


The  specific  resistance  of  lead  is  not  far  from  that  of  average 
German  silver.  It  is  about  twice  that  of  iron  and  about  twelve 
times  that  of  copper.  The  remarkably  large  skin  effect  indi- 
cated for  iron  by  this  table  is  due  to  its  relatively  large  magnetic 
permeability.  The  tabular  values  should  be  proportional  to 
the  square  roots  of  the  specific  resistances  of  the  different  metals 
and  inversely  proportional  to  the  square  roots  of  their  mag- 
netic permeabilities.  For  metals  of  a given  specific  resistance, 
the  tabular  values  should  be  inversely  proportional  to  the 
square  roots  of  the  frequencies.  In  view  of  the  high  frequency 
harmonics  that  it  is  desirable  to  preserve  in  telephone  currents, 
the  table  indicates  a reason  why  iron  should  be  rejected  as  a 
material  for  telephone  conductors.  It  must  be  understood  that 
the  permeability  of  iron  wires  depends  upon  the  quality  of  the 
iron  and  the  magnetic  density  set  up  in  its  mass,  and  that 
it  may  vary  from  a few  tens  of  units  to  many  hundreds  of 
units.  The  tabular  value  of  300  is  taken  as  a median  or  rep- 
resentative figure  for  iron  subjected  to  rather  low  magnetiz- 
ing forces. 

The  formula  for  the  self-inductance  of  a cylindrical  wire  was 
developed  in  the  preceding  article  on  the  assumption  of  a uni- 
form distribution  of  current  over  the  cross  section  of  the  con- 
ductor. The  disturbance  of  this  distribution  by  skin  effect 
reduces  the  self-inductance,  the  formula  for  self-inductance  in 
henrys  per  centimeter  of  length  becoming 


La  = 

Vlog^  + /“' 


1 1 ( 2 tt/^Y  | 13 


2 tt/aMY 


_2  48\  109p  J 8640V  109p  J 


f 2 7T 

V 1( 


— etc. 


or 


La=L-/ 


■ 1 f 2 wffi'AV  _ _13_  ( 2 vfu'AY 


_48V  10 9p  J 8640  V 109p  J 


+ etc. 


lO-9, 


10-9. 


The  limiting  values  of  Ra  and  La  are  Ra  = R and  La  = L for 
steady  currents  or  alternating  currents  of  low  frequency  in  con- 
ductors of  moderate  thickness;  and  Ra  = cc  and  La  — 2 log  - for 
very  high  frequencies. 

213.  Mutual  Induction  of  Parallel  Distributing  Circuits.  — 

Where  two  or  more  electric  light  or  power  circuits  carrying 
alternating  currents  run  parallel  to  each  other,  they  act  in- 
ductively upon  each  other,  and  in  some  cases  the  mutual  indue- 


912 


ALTERNATING  CURRENTS 


tion  may  cause  considerable  interference  with  the  uniformity 
of  the  voltage  on  the  lines.  The  mutual  inductance  of  any  two 
parallel  circuits  of  indefinitely  great  length  may  be  easily  cal- 
culated, provided  the  distances  apart  of  - the  different  wires 
composing  the  circuits  are  known.  The  mutual  inductance  of 
the  two  circuits  given  in  henrys  is  equal  to  10-8  times  the 
number  of  lines  of  force  which  pass  through  or  link  with  one 
of  the  circuits  due  to  one  ampere  flowing  in  the  other  circuit, 
provided  there  is  no  iron  in  the  magnetic  path  ; and  this  num- 
ber of  lines  of  force  is  equal  to  the  algebraic  sum  of  the  number 
of  lines  of  force  embraced  by  the  first  circuit  which  would 
be  set  up  by  the  current  in  the  individual  conductors  of  the 
second  circuit  taken  separately.  The  method  of  Art.  211  is 
therefore  directly  applicable  to  the  calculation  of  the  mutual 
inductance  of  two  long  and  parallel,  narrow  circuits.  The 
following  examples  represent  the  commonest  arrangements  of 
circuits  on  pole  lines. 

Suppose  that  a,  a'  and  5,  b'  represent  the  conductors  of  two 
circuits,  and  that  the  order  of  the  wires  is  a — a'  — b — b' , the  dis- 
tance apart  center  to  center  of  the  wires  of  circuit  A is  x,  of 
circuit  B is  y,  and  of  the  adjacent  wires  of  the  two  circuits 
( a ' — b~)  is  z ; then  if  we  consider  the  currents  as  concentrated  at 
the  centers  of  the  wires,  which  makes  but  an  insignificant  error 
with  the  ordinary  dimensions  of  conductors  and  circuits,  and 
consider  the  space  between  two  planes  perpendicular  to  the  cir- 
cuits and  one  centimeter  apart,  the  number  of  lines  of  force 
due  to  a current  of  one  ampere  in  a',  which  pass  through  the 
circuit  B between  the  planes,  is  (Art.  211) 


and  the  number  of  lines  of  force  due  to  a current  of  one 
ampere  in  a which  pass  through  the  circuit  B between  the 


planes,  is 


C*+y+z  2 da 
Uz  10  a 


The  total  number  of  lines  of  force  set  up  by  the  current  of  one 
ampere  in  circuit  A , which  pass  through  the  circuit  B between 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


913 


the  planes,  is  <E>0  4-  4>a<,  the  number  which  link  through  the  B 
circuit  in  a length  of  l centimeters  is 

Z($a+ 

and  the  mutual  inductance  of  the  parallel  circuits  of  length  l 
centimeters  is 

M=  1(4'.  + K..)  = 2 l A <r+£  _ j x + g + A 
108  109V  8 a * x + z J 

_ ~lL  Iqp-  Q + *)0/  + z)  = 4-60  log  (x  + zYy  + z) 
109  c z(x-\-y  + z)  109  5,10  z(x  + y + z) 


If  x = ?/,  this  becomes 


■,  (x  + z')2  4.60  Z.  (x+z')2 

log-  = Tor  l0*i»a(2UjTj ; 


and  if  x = y = z,  it  becomes 


jfff-  4- 60  ^ i 4 
- IQ9  1Ogl03 


575  l 
109  ’ 


where  l is  the  length  of  the  parallel  circuits  in  centimeters. 

The  mutual  inductance  in  millihenrys  per  1000  feet  of  dis- 
tance in  which  the  circuits  are  parallel  is,  using  common  loga- 
rithms, 


M=  .1404  log 


M"  --  .01752. 


(x  + z)(y  + z)  . 
z(x  + y + z) 


M'  = . 1404  log  -(x  + z')2- ; 

° z(2  x + z) 


Exchanging  the  order  of  the  wires  so  that  circuit  A is  be- 
tween the  conductors  of  circuit  B,  thus  b — a — a'  — b',  changes 
the  formulas.  Here  the  algebraic  sum  of  the  number  of  lines 
of  force  set  up  by  the  circuit  A which  link  with  circuit  B is 
equal  to  the  total  number  of  lines  of  force  set  up  by  circuit  A 
minus  the  number  passing  backwards  through  b — a and  a'  — b'. 
Suppose  that  distance  a — a'  is  equal  to  x and  distances  b — a 
and  a ' — b'  are  each  equal  to  y , then  the  total  number  of  lines 
of  force  set  up  by  one  ampere  in  a length  of  one  centimeter  of 
circuit  A , is 

<D0=.4  log  - 4-  .2, 

r 

3 N 


914 


ALTERNATING  CURRENTS 


(r  being  the  radius  of  the  conductor),  and  the  number  of  lines 
due  to  one  ampere  in  circuit  A,  which  pass  between  the  planes 
through  the  space  b — a,  is 


= .2  log,  V-  + .1  - .2  log,  _ .2  log. 


r(x  + y') 


+ -1  •> 


xy 


r(x  + y) 


ii 

109 


x -f  y _ 9.20  l 
y 109  --  y 


TEtz-i  JU  ~ If  V I X ~\-  If 

- ^ log,  — ^ = -T^r  log10 


T,  Tvrn  9.20 1,  0 2.77  l 

If  x = y,  M"=  — — log102  = - 


109 


The  mutual  inductance  in  millihenrys  per  1000  feet  of  dis- 
tance in  which  the  circuits  are  parallel  is,  using  common  loga- 
rithms, 

M’=  .2807  log10^hJ/;  M"  = .08451. 

y 

If  the  circuits  are  not  in  the  same  plane,  as,  for  instance, 
they  are  arranged  thus, 

a — a ’ , 
b-b', 


and  the  distance  a — a'  is  x,  the  distance  b — b'  is  y , the  dis- 
tance a'  — b'  is  z , the  distance  a'  — b is  w,  a — b is  v,  and  a — b' 
is  u ; then  the  formulas  are 


d>a  = • 2 flog,  - — log,  -'W  .2  log,  - , 
\ r r)  v 


4 >«•  = - 2(  l°g,^  - log,f)  = -2  lo£ 


w 


and 


nr  ^ i \ 2 Z,  uiv  4.60  £ , uw 

M=  Tua  (4>“  + *“■>  = 105  ‘°g-  ^ l0*i.  T7 ' 


If  one  circuit  is  directly  beneath  the  other  and  x = y,  v = z. 
then  w = u = V.t2  + z2,  and  the  formula  becomes 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


915 


The  mutual  inductance  in  millihenrys  per  1000  feet  of  dis- 
tance in  which  the  circuits  are  parallel  is,  using  common  loga- 
rithms, 

M=  .1403  log  — ; M'  = .1403  log  ; M"  = .04225. 

vz  Zl 


In  case  the  arrangement  is 


a — b 
V -a! 


, one  circuit  being  directly 


above  the  other  and  the  distance  a — b equal  to  the  distance 
b'  — a,  that  is,  the  wires  are  on  the  corners  of  a square  with  the 
plane  of  each  circuit  tracing  the  diagonal,  then  the  flux  set  up 
by  either  circuit  is  tangential  with  the  other  and  the  mutual 
inductance  is  zero,  there  being  no  linkages. 

These  results  plainly  show  that  the  mutual  inductance  of  two 
circuits  is  entirely  independent  of  the  actual  distances  apart  of 
the  conductors  composing  the  circuits,  but  depends  wholly 
upon  the  relative  values  of  the  distances.  The  mutual  induct- 
ance of  two  circuits  is  a maximum  when  the  circuits  are  exactly 
superposed,  in  which  case  M=VL'L"=L,  and  decreases  as 
the  distance  between  the  circuits  is  increased  in  comparison 
with  the  distance  apart  of  the  conductors  of  each  circuit  ; con- 
sequently, mutual  inductance  between  circuits  on  the  same 
pole  line  may  be  reduced  by  decreasing  the  distance  apart  of 
the  conductors  of  each  circuit  and  increasing  the  distance  apart 
of  the  circuits.  A better  way  to  avoid  annoyance  from  the 
effects  of  mutual  inductance  in  most  cases  is  to  transpose  the 
positions  of  the  circuits  with  reference  to  each  other  at  fixed  in- 
tervals of  distance  on  the  pole  line,  so  that  the  inductive  effects 
of  the  circuits  on  each  other  are  in  opposition  in  different  equal 
parts  of  the  line,  and  neutralize  each  other  for  the  line  as  a whole. 
If  there  are  more  than  two  circuits  on  a pole  line  and  trans- 
positions are  needed,  it  is  important  to  make  the  transpositions 
so  that  the  magnetic  mutual  reactions  which  occur  between  any 
of  the  circuits  all  balance  off  and  neutralize.  For  this  purpose 
it  is  necessary  that  the  transpositions  shall  be  made  properly. 
In  any  transposition  interval,  one  line  may  be  run  straight 
through,  one  should  be  transposed  at  the  middle,  one  at  a 
quarter  way  from  each  end  of  the  interval,  one  at  the  quarters 
and  the  middle,  etc. 

The  effect  of  mutual  induction  between  two  circuits  is  to  set 
up  a voltage  in  one  when  the  current  in  the  other  varies.  If 


916 


ALTERNATING  CURRENTS 


the  current  is  a sinusoidal  alternating  one,  this  voltage  is  (Arts. 
119  and  120)  2 7 rfMI,  and  the  effect  of  an  alternating  current 
in  one  circuit  upon  another  circuit  is  easily  determined  if  Afis 
known.  When  the  two  circuits  are  fed  from  the  same  single- 
phase alternator,  the  induction  of  one  upon  the  other  is  in  quad- 
rature with  the  current  in  the  first,  and  the  phase  relation  of  the 
voltage  induced  in  the  second  relative  to  the  impressed  voltage 
therein  depends  on  the  current  lag  in  the  first.  If  this  is  zero, 
the  induced  and  impressed  voltages  are  in  quadrature,  while 
they  are  in  opposition  if  the  lag  is  90°.  The  result  is  a dis- 
placement of  the  voltage  waves  and  a drop  of  voltage  along  the 
lines.  The  effect  of  reversing  the  connection  of  one  of  the  cir- 
cuits to  the  alternator  causes  an  interchange  of  the  voltage 
relations  of  the  two  circuits,  but  nothing  else. 

If  the  circuits  are  fed  from  different  alternators  the  frequen- 
cies of  which  are  slightly  different,  the  inductive  voltage  and 
impressed  voltage  interfere  in  each  circuit  so  as  to  form  pulsa- 
tions or  beats,  the  frequency  of  which  is  equal  to  the  difference 
of  the  two  alternator  frequencies,  and  the  amplitude  of  which 
is  the  sum  of  the  impressed  voltage  and  the  induced  voltage. 
This  may  cause  a perceptible  winking  of  incandescent  lamps 
connected  to  mutually  inductive  circuits  carrying  heavy  cur- 
rents of  nearly  the  same  frequency. 

214.  Effects  of  Self  and  Mutual  Induction  in  Polyphase  Cir- 
cuits. — The  effects  of  self  and  mutual  induction  in  polyphase 
circuits  may  be  determined  by  using  the  principles  set  forth 
in  the  preceding  articles,  provided  the  resultant  effect  of  the 
differing  phases  is  always  considered.  In  unbalanced  circuits, 
if  mutual  inductive  effects  are  allowed  to  occur,  they  may  in- 
crease the  defect  in  balance.  Mutual  induction  between  the 
phases  of  one  polyphase  feeder  may  be  avoided  by  suitably 
placing  the  conductors.  Thus,  in  the  case  of  a four-wire  two  ' 
phase  feeder,  the  wires  may  he  located  on  two  cross  arms  so 
that  the  planes  of  the  conductors  of  the  two  phases  form  the 

diagonals  of  a square,  thus  a where  a , a are  the  conduc- 

o a 

tors  of  one  phase  and  5,  b are  the  conductors  of  the  other.  In 
this  case  there  is  no  mutual  induction  between  the  circuits. 
When  this  arrangement  cannot  be  utilized  with  quarter  phase 
lines,  transpositions  must  be  resorted  to. 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


91T 


Three-phase  circuits  may  he  erected  on  insulators  located  so 
that  the  three  wires  occupy  the  corners  of  an  equilateral 

triangle,  thus  a . The  lines  of  force  set  up  by  the  current 
c b 

in  any  one  of  the  conductors  are  circles  surrounding  the  conduc- 
tor and  are  therefore  tangent  to  the  plane  of  the  loop  composed 
of  the  other  two  conductors.  Hence  they  do  not  link  the  loop 
and  no  mutual  induction  occurs.  When  this  arrangement  can- 
not be  utilized,  transpositions  may  be  used  to  neutralize  mutual 
inductance. 

When  two  or  more  feeders  are  near  each  other,  as  on  the 
same  line  of  poles  or  towers,  mutual  induction  between  feeders 
may  be  neutralized  by  transposing  the  wires  of  each  feeder 
spirally  in  such  a way  that  the  inductive  actions  in  successive 
lengths  balance  each  other.  With  the  equilateral  triangle  ar- 
rangement of  three-phase  conductors,  the  spiraling  of  each 
feeder  composed  of  three  conductors  may  be  carried  out  with 
regard  onty  to  the  effects  of  the  feeders  on  each  other  and  on 
other  neighboring  circuits;  but  when  the  equilateral  triangle 
arrangement  is  not  used,  the  spiraling  of  each  feeder  should 
be  accompanied  by  suitable  transpositions  to  neutralize  the 
mutual  induction  occurring  between  the  phases  of  the  feeder. 

The  vector  relations  of  the  voltages  in  four-wire  two-phase 
circuits  when  the  conductors  lie  in  a plane  (as  when  they  are 
mounted  on  the  pins  of  the  same  cross  arms  on  a pole  line 
or  are  supported  one  above  another  by  suspension  insulators) 
are  illustrated  in  Fig.  519.  The  assumed  relative  positions  of 
the  wires  are  illustrated  in  the  upper  part  of  the  figure,  at  AB, 
and  the  vector  relations  in  the  left  hand  and  lower  parts.  The 
vector  diagram  is  drawn  for  conditions  of  balanced  load  at  the 
receiver  end  of  the  line. 

In  the  vector  diagram  of  Fig.  519,  0EA  and  0EB  are  the  vol- 
tages at  the  receiver  in  the  respective  phases  90°  apart,  and 
OIA  and  OIB  are  the  corresponding  currents.  The  voltage  .at 
the  generator  0E°  in  each  circuit  is  equal  to  the  vector  sum  of 
the  voltage  at  the  receiver  and  components  numerically  equal 
and  opposite  respectively  to  the  IR  drop  in  the  circuit  E' E, 
the  IX  drop  in  the  circuit  E" E' , and  the  voltage  E°E"  intro- 
duced in  the  circuit  by  the  inductive  influence  of  the  current  in  the 
other  branch  of  the  two-phase  circuit.  The  generator  voltages 


918 


ALTERNATING  CURRENTS 


in  the  two  circuits  are  therefore  0EA  and  0EB°.  The  effect  of 
the  mutually  induced  voltage  is  therefore  to  increase  the  numeri- 
cal difference  and  decrease  the  angular  difference  between  the 
genei’ator  voltage  and  the  voltage  delivered  to  the  receiver  in 

E°* 


Fig.  519.  — Diagram  of  Vector  Voltage  Relations  in  Four-wire,  Four-phase  Circuits 
when  the  Four  Wires  lie  in  a Plane,  as  at  AB. 

the  leading  phase,  and  to  decrease  the  numerical  difference  and 
increase  the  angular  difference  between  the  generator  voltage 
and  the  voltage  delivered  to  the  receiver  in  the  lagging  phase, 
compared  with  the  relations  when  mutual  induction  is  absent. 

If  Ea°  is  the  line  voltage  at  the  generator  for  the  leading 
phase,  Ea  the  line  voltage  at  the  receiver,  R A,  L<  and  JSI the  line 
resistance,  self-inductance,  and  mutual  inductance,  IA  the  cur- 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


919 


rent,  and  Z , R.  and  X the  impedance,  resistance,  and  reactance 
of  the  load  in  the  same  phase,  then 

Ia=~z  = ir+jx  and 

Also 

Ea  = EA°  - IARA  - jcoLlA  -jcoMTs 

P o BaEa(R  — jX')  . coLEa (R  — jX)  — <bMEa (X  +jR) 

A i22  + X2  3 R2  + X2 


J R +jX 


= xA0-xA 

Hence 

ea°=ea 


RRa  + XXA  + coMR  . RXa  - XRa  - a)i¥X1 
—+J  — 


R?  + X2 


R2  + X2 


' RRa  + XX,  + CoMR  . RXi  - XRa  - coMX ' 

1+  TY>.  . + J ~ 


R2  + X2 


R2  + X2 


By  the  same  process 


eb°  = eb 


■ RRa  + XX A — (o3IR  ,RXa-XRa  + coMX 
1+  R2  + X2  +J  R2  + X2 


When  the  load  is  non-reactive,  X = 0 and  the  equations  for 
E ° and  Eb°  become 

ea° =ea(  i + 


e°  = eb  1 + 


R,  — co3f  . ,X4 


A 


R 


It  may  be  observed  from  the  equations  as  well  as  the  diagram 
that,  when  mutual  induction  exists  between  the  phases,  the 
generator  voltage  is  not  balanced  in  case  the  voltage  and  cur- 
rent  at  the  load  are  balanced  ; and  conversely,  if  the  generator 
voltage  is  balanced,  it  is  necessary  to  abate  mutual  induction 
or  else  introduce  special  phase  regulators  in  order  to  obtain 
balanced  voltage  conditions  at  the  load.  Inasmuch  as  the  IZ 
drop  of  a distribution  line  is  ordinarily  much  greater  than  the 
mutually  induced  voltage  between  circuits  on  the  same  poles, 
the  unbalancing  by  mutual  induction  may  not  be  very  serious. 

Figure  520  shows  the  effect  of  mutual  induction  in  a three- 
phase  circuit  with  three  wires  in  one  plane,  as  when  they  are 
erected  on  the  same  cross  arm  on  each  pole  of  a pole  line,  or 
are  suspended  in  a vertical  plane  by  suspension  insulators. 
The  lines  OEab , OEbc , and  OEca  are  balanced  delta  line  voltages 


920 


ALTERNATING  CURRENTS 


at  the  receiver,  OEa,  OEb,  and  OEc  representing  the  correspond- 
ing wye  voltages  from  each  line  conductor  to  the  neutral  point, 
and  OIa , OIb , and  OIc  are  the  line  currents.  In  this  case  the 
mutual  inductance  of  the  middle  conductor  b on  the  loop  com- 
posed of  conductors  a and  c is  zero,  provided  the  distance 
between  a and  b is  equal  to  the  distance  between  b and  c , since 


Fig.  520.  — Diagram  for  showing  the  Effect  of  Mutual  Induction  in  a Three-phase 
Circuit  when  the  Wires  are  in  One  Plane,  as  at  abc. 


under  those  circumstances  the  flux  set  up  through  the  loop  a — c 
by  current  in  b is  half  upward  and  half  downward.  Conductor 
c,  however,  possesses  mutual  inductance  with  respect  to  the  loop 
a — b,  and  conductor  a has  mutual  inductance  with  respect  to 
loop  b — c.  In  case  conductor  b is  not  exactly  halfway  between 
a and  c,  mutual  inductance  also  exists  between  conductor  b and 
loop  a — c. 

In  a balanced  three-phase  circuit  the  mutual  induced  voltage 
set  up  in  a loop  of  two  conductors,  such  as  a and  b , by  current 
in  the  third  conductor  is 

-y  V 2 2 7 rfMIL  sin  [a  + 120°  - (30°  + 0)], 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


921 


when  the  instantaneous  delta  voltage  on  the  loop  is  propor- 
tional to  sin  a.  Therefore,  if  9 = 0,  that  is,  the  receiver  cur- 
rent is  in  phase  with  the  wye  voltage  at  the  receiver,  the 
mutually  induced  voltage  is  in  the  same  phase  as  the  delta 
voltage  at  the  receiver.  If  9 is  not  equal  to  zero,  the  mutually 
induced  voltage  differs  in  phase  from  the  delta  voltage  at  the 
receiver  by  the  angle  9. 

If,  in  the  arrangement  of  conductor's  illustrated  in  the  left- 
hand  part  of  Fig.  520,  the  conductor  b is  moved  perpendicular 
with  respect  to  the  plane  of  loop  a — e,  tire  mutual  inductance  of 
a with  respect  to  b — c and  of  c with  respect  to  a — b decreases 
until  b has  reached  a point  at  which  distances  a — b = b — c = c—  a. 
Under  the  latter  circumstances,  the  conductors  occupy  positions 
corresponding  to  the  corners  of  an  equilateral  triangle,  the  lines 
of  force  due  to  any  one  conductor  are  tangent  to  the  plane  of 
the  loop  composed  of  the  other  two,  and  the  mutual  inductance 
is  zero.  If  conductor  b is  moved  farther  from  the  a — c plane, 
the  mutual  inductance  again  increases,  but  the  direction  of  the 
fluxes  of  a and  a through  the  loops  b — c and  a — b are  reversed, 
so  that  the  mutually  induced  voltages  are  reversed.  This  is 
obvious  from  the  following  considerations:  If  distance  a — c is 
#,  distance  a — b is  y,  and  distance  b — e is  z,  then  the  mutual 
inductance  of  conductor  a on  loop  b — c is 


9/  T 

Ma  = ^logA 

10M  y 

and  the  mutual  inductance  of  conductor  c on  loop  a — b is 


Mc  = p-\oge-. 
c 109  Sez 


When  b is  nearer  the  plane  of  a — c than  (distance  a — <?)  x sin  60°, 
log  - and  log£  - are  positive  and  therefore  Ma  and  Mc  are  posi- 

ey  z 

QC  jC 

tive;  but  when  b is  farther  from  the  plane,  loge  - and  loge  - are 

V z 

negative  and  Ma  and  Mc  therefore  become  negative. 

In  Fig.  520,  EE'  for  each  branch  of  the  diagram  represents 
the  line  resistance  and  reactance  drop  and  E'EQ  represents  the 
mutually  induced  voltage,  when  the  receiver  voltages  and  cur- 
rents are  as  shown.  The  vectors  OE^0,  OEb° , and  OE c°. 


922 


ALTERNATING  CURRENTS 


therefore,  represent  the  delta  voltages  at  the  generator.  In 
this  illustration  the  distance  between  conductors  a and  b is 
taken  equal  to  the  distance  between  conductors  b and  c,  the 
three  conductors  being  in  a plane,  and  there  is  no  mutual  vol- 
tage in  the  c—  a loop.  It  will  thus  be  observed  that  the  voltages 
at  the  generator  end  of  the  line  must  be  unbalanced  in  case  the 
voltages  at  the  receiver  end  are  to  remain  balanced,  unless 
the  conductors  can  be  located  on  the  apices  of  an  equilateral 
triangle  or  transpositions  are  resorted  to. 

215.  Electrostatic  Capacity  of  Parallel  Conductors.  — When 
the  voltage  used  for  electrical  transmission  of  power  is  high,  the 
amount  of  transmitted  current  is  relatively  small,  especially 
when  the  load  is  light,  and  the  reactance  drop  in  the  line  may 
be  largely  affected  by  the  electrostatic  capacity  between  the  con- 
ductors. It  is,  therefore,  necessary  to  investigate  the  magni- 
tude of  the  electrostatic  capacity  between  parallel  conductors. 

Electrostatic  capacity  of  two  neighboring  conductors  with 
respect  to  each  other  may  be  defined  as  the  ratio  of  the  electro- 
static charge  on  eacli  to  the  difference  of  potential  between  the 
conductors  which  is  set  up  by  the  charges,  one  of  which  charges 
is  positive  and  the  other  of  which  is  equal  numerically  but  of 
opposite  kind.  The  capacity  in  farads  is  the  ratio  between  the 
charge  measured  in  coulombs  and  the  difference  of  potential 
measured  in  volts.  When  the  conductors  are  large  parallel 
plates  whose  distance  apart  is  small  compared  with  their  super- 
ficial dimensions,  the  capacity  may  be  readily  found  on  the 
assumption  that  a charge  distributes  uniformly  over  them, 
which  is  true  except  for  portions  near  the  edges.  These  con- 
ductors being  charged,  and  away  from  the  influence  of  other 
conductors,  one  carries  a positive  charge  and  the  other  carries 
an  equal  negative  charge.  The  electrostatic  field  between  them 
is  uniform  except  near  the  edges,  and  the  difference  of  poten- 
tial measured  from  one  to  the  other  is  equal  to  the  intensity  of 
the  field  multiplied  by  the  centimeters  of  distance  between  the 
plates.  The  measure  of  the  intensity  of  the  field  is  the  force 
in  dynes  which  would  be  experienced  by  a C.  G.  S.  electrostatic 
unit  of  charge  moving  from  the  plate  of  lower  potential  to  the 
plate  of  higher  potential. 

The  force  which  measures  the  field  intensity  is  numerically 
equal  to  1 /K  times  the  electrostatic  lines  of  force  per  square 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


923 


centimeter  in  the  dielectric  (measured  in  a plane  perpendicular 
to  the  flux)  ; and  this  flux  is  equal  to  4 7 rcr,  in  which  a is  the 
electrostatic  charge  per  square  centimeter  on  each  plate,  meas- 
ured in  electrostatic  units,  and  K is  the  dielectric  constant  of 
the  insulator  lying  between  the  plates.  Also  Q = a A where  A 
is  the  area  of  each  plate  in  square  centimeters.  Therefore,  by 
definition,  the  capacity  of  the  condenser  composed  of  the  par- 
allel plates  is,  in  electrostatic  units  of  measure, 

C-Qa-  Qa  - KA_ 

a Ea  Fad  4 7 rdaa  4 ird 

To  transform  from  C.  G.  S.  electrostatic  units  to  C.  G.  S. 
electro-magnetic  units,  with  which  the  practical  units  conform, 
it  is  necessary  to  divide  by  v 2 where  v is  the  velocity  of  light, 
and  to  transform  from  C.  G.  S.  electro-magnetic  units  to  farads 
it  is  necessary  to  multiply  by  109  since  a C.  G.  S.  electro-mag- 
netic unit  of  capacity  is  TO9  times  as  large  as  a farad.  The 
value  of  v has  been  experimentally  proved  to  be  substantially 
3 x 1010.  Consequently,  the  capacity  of  the  parallel  plates  in 
farads  is 

n__KA  10-9 
1.13  d ’ 

A being  the  area  of  each  plate  in  square  centimeters  and  d the 
distance  between  them  measured  in  centimeters.  For  inch 

KA  IO-9 

measure  the  formula  is  C=  2.25— . 

d 

In  the  case  of  concentric  cylinders  of  great  length  compared 
with  the  radial  clearance  between  them,  one  of  which  carries  a 
positive  charge  and  the  other  an  equal  negative  charge,  the 
surface  density  of  the  charge  is  uniform  on  each  cylinder 
except  close  to  the  ends.  Considering  the  end  effects  negligi- 
ble in  long  cylinders,  the  lines  of  force  in  the  electrostatic  field 
are  radial.  Inasmuch  as  the  radial  force  exerted  on  an  electro- 
static unit  charge  at  any  point  P outside  of  a uniformly 
charged  cylinder  of  indefinitely  great  length  is 

~ dynes, 

Kr 

where  q is  the  charge  in  C.  G.  S.  electrostatic  units  per  centi- 
meter of  length  of  the  cylinder  and  r is  the  radial  distance  in 


924 


ALTERNATING  CURRENTS 


centimeters  from  the  axis  of  the  cylinder  to  the  point  P ; * and 
also,  inasmuch  as  the  force  inside  of  a uniformly  charged 
cylinder  is  zero,  when  the  influence  of  the  ends  may  be 
neglected ; therefore  the  capacity  in  C.  G.  S.  electrostatic 
units  of  the  concentric  cylinders  is 


C -<?«-  & - KI<1  = Kl 

frFadr  2 qf  ^ 2l0gX 

*/r  r rj 

in  which  and  r2  are  respectively  the  outer  radius  of  the  inner 
cylinder  and  the  inner  radius  of  the  outer  cylinder,  and  l is 
the  length  of  each  cylinder  in  centimeters.  Reducing  the 
equation  to  represent  capacity  in  farads  gives 


C - 


Kl  10-9 
1800  logE  - 


and,  finally,  reducing  the  equation  to  represent  farads  per  1000 
feet  of  each  cylinder  and  using  common  logarithms  gives 


r _ 7.35  K10~9 

logw* 

ri 

When  the  cylinders  are  not  concentric  but  the  distance  be- 
tween centers  is  a centimeters,  one  cylinder  being  entirely  sur- 
rounded by  the  other,  the  last  equation  becomes 


2 r2r 


7.35  W10-9 


+ r,2  — a2\ 
2 r r J 

* }2ri  ' 


These  formulas  apply  to  an  underground  insulated  conductor 
surrounded  by  a lead  sheath,  in  the  usual  construction  of  high 
voltage  cables  for  underground  use. 


* The  force  at  P due  to  the  charge  on  any  element  of  length  of  the  cylinder 

is  Qf'1 — — . Only  the  radial  component  of  this  is  effective,  as  components 

parallel  to  the  cylinder  neutralize  each  other,  and  the  radial  component  is 

2^ sin  0 = - — x — = The  force  due  t0  tjie 

K(x>  + r2)  K(x2  + r2)  Vx2+r2  jv(.r2+r2)i 

entire  cylinder  of  indefinitely  great  length  is  therefore 

—7^ dynes. 

K(x2  + r2)2  Kr 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


925 


In  case  the  cylinders  are  side  by  side  instead  of  one  inside 
the  other,  as  in  the  case  of  two  overhead  line  wires  or  two 
separately  insulated  conductors  lying  side  by  side  in  the  same 
underground  cable,  the  equation  takes  a different  form.  The 
following  development  of  the  equation  is  simple. 

Considering  two  indefinitely  long  parallel  conducting  fila- 
ments A and  A!  (perpendicular  to  the  plane  of  the  page),  Fig. 
521,  charged  respectively  with  + q and  — q electrostatic  units  of 
charge  per  centimeter  of  length,  the  force  experienced  by  a 
unit  charge  at  any  point  P in  the  electrostatic  field  of  filament 


9 

A and  distant  r centimeters  from  the  filament  would  be  if 

Kr 


filament  A'  were  removed,  and  the  potential  at  the  point  due 
to  A is 


=f 


if 


Also,  the  potential  at  P due  to  A'  alone  is 


'•--JTfc*' 


if  r'  is  the  radial  distance  of  the  point  from  the  filament  A' . 
The  total  potential  at  P due  to  the  charges  on  A and  A ' is 
therefore 


vp=  V+  V'  = 


dr 


2 q-.  r' 

=iclos-y 


Therefore,  if  the  point  P moves,  the  potential  VP  at  the 
point  P does  not  change  provided  the  locus  of  P maintains  the 

ratio  — constant ; and  this  locus  is  a circle  with  its  center  0 
r 

(Fig.  521)  located  on  the  line  joining  A and  A'  extended  and 
with  radius  of  length  a which  is  a mean  proportional  between 
the  lengths  OA  and  OA' . If  P1  is  the  intersection  of  the 
circle  with  the  line  OA',  these  conditions  arise, 


A'P 


AP 


- m, 


OP , _ OA' 
OA  OP  i 


92G 


ALTERNATING  CURRENTS 


in  which  P is  any  point  on  the  circle,  P,  is  the  point  where  the 
circle  intersects  the  line  OA! , and  m is  a constant.  It  therefore 
follows  that 

OP  OP,  _ OA'  A'P,  A'P 

OA  OA  OP,  AP,  AP  m' 

and  by  division  in  the  third  and  fourth  ratios, 

OP  = A-  P\  x AP, 

1 A'P , - AP, 

The  equipotential  surfaces  around  each  one  of  the  charged 
filaments  are  cylinders  of  radius  increasing  from  zero  to  infinity 


Fig.  521.  — Diagram  showing  a Locus  of  Equal  Potential  between  Two  Parallel 
Charged  Filaments  A and  A'. 


as  the  point  P is  taken  at  increasing  distances  from  the  fila- 
ment. These  merge  into  a plane  of  zero  potential  when 
A'P,  = APV  P,  being  halfway  between  the  two  equally  but 
oppositely  charged  filaments.  This  plane  touches  infinity  and 
has  the  same  potential  (namely,  zero)  as  space  at  infinite  dis- 
tance from  the  charges. 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


927 


Introducing  a cylinder  of  conducting  material  with  its  outer 
surface  coincident  with  the  position  of  any  one  of  the  equipo- 
tential  surfaces  will  not  alter  the  electrostatic  field;  and  the 
electrostatic  field  of  two  charged  cylindrical  parallel  metal  con- 
ductors may  be  computed  by  considering  the  surfaces  of  each 
as  an  equipotential  surface  surrounding  a charged  filament,  the 


Fig.  522. — Diagram  for  illustrating  Method  of  finding  the  Mutual  Potentials  of  Two 
Charged,  Cylindrical,  Parallel  Conductors. 


center  to  center  distance  between  tbe  cylinders  being  the 
center  to  center  distance  00'  between  the  corresponding  equi- 
potential circles  (Fig.  522).  Then,  according  to  the  preceding 
paragraphs,  the  mutual  potential  at  the  surface  of  cylinder  A is 


D=^iog.r,, 

and  the  mutual  potential  at  the  surface  of  the  cylinder  A'  is 


7 


Therefore,  the  difference  of  potential  in  C.  G.  S.  electrostatic 
units  is 


K = vA  - VA,  = ii  log  r*  = log, 
K r,r,  K 


aa 


1 4 


since  the  relations  proved  by  Fig.  521  are  in  Fig.  522, 


r2  _ A'P  __  OP  a d r„  _ AP'  _ O' P'  _ a ' 
rx  AP  OA  «:UU  r4  A’P'  O' A!  ar 


The  electrostatic  capacity  with  respect  to  each  other  of  the  two 


928 


ALTERNATING  CURRENTS 


parallel  cylindrical  conductors  given  in  C.  G.  S.  electrostatic 
units  is  therefore 


o.  = i 


Kl 


2 l0ge 




aa 

aa' 


where  l is  the  length  of  each  cylinder  in  centimeters. 

It  is  desirable  to  convert  this  formula  into  one  involving 
only  the  radii  a and  a'  and  the  center  to  center  distance  3 of 
the  cylinders,  by  eliminating  the  distances  a and  This  may 
be  easily  done,  since  a + «'  -f-  d = 3.  Then  putting 


gives 

Whence 


32  - a2-  (V)2_  t 


aa 


— = b±  V62  - 1. 

aa' 


ca= 


Kl 


21oge  (b  ± Vi2  — 1) 


When  the  cylinders  are  eccentric,  as  in  the  case  under  consider- 
ation, the  positive  sign  applies  before  the  radical  in  the  denom- 
inator of  the  last  expression. 

To  bring  this  to  farads  per  1000  feet  of  conductor  in  the 
circuit  and  using  common  logarithms,  this  must  be  multiplied 
by  7.35  x 10-9,  so  that 

C 7.35  K 1Q~9  _ 3.68  K 10~9 

1 21og10(6  +V62-  1)  log10(6  + VP-  1) 

When  the  two  cylinders  are  of  the  same  diameter,  a = a', 
whence  b = ^ — 1, 


and 

Therefore 


b + VJ2  - 


3.68  Kl 0-9 


1.84  K 10~9 


lo 


olO 


+ 


z a 


m-')' 


When  the  distance  apart  (center  to  center)  of  the  conductors 
is  large  compared  with  the  diameter  of  each  conductor,  as  for 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


929 


instance  if  this  ratio  is  as  great  as  is  usual  with  overhead  line 
ires,  \j\ 


wir 


^ — 1 approximates  to  and  a closely  ap- 
proximate formula  for  the  capacity  in  farads  per  1000  feet  of 
conductor  may  be  written 


^2  = 


1.84  K 10-9 


log 


10' 


When  the  conductors  are  in  air,  the  dielectric  constant  is 
unity ; therefore  for  overhead  line  wires  the  capacity  in  farads, 
measured  from  conductor  to  conductor,  per  thousand  feet  of 
conductor,  is  usually  obtained  from  the  expression 


^2  = 


1.84  x 10~9 

i 3 

log„- 


Since  log6  (b  + v/62  — 1)  = cosh-1  b,*  the  exact  formula  for 
the  capacity  in  farads  per  thousand  feet  of  conductor  may  be 
written 

c = 16.94  iT10-9 
1 cosh-1 b 

which  is  easily  computed  with  the  aid  of  a table  of  the  functions 
of  hyperbolic  trigonometry,  such  as  are  contained  in  the  Smith- 

,32 

sonian  Mathematical  Tables.  In  this  formula  b = — - — 1 when 

2 a2 

the  conductors  are  of  equal  diameters,  in  which  case,  putting 
c for  the  ratio  of  the  center  to  center  distance  to  the  diameter 

Q 

of  a conductor,  c = — and  b = 2 e2  — 1. 

2 a 

Therefore,  b + ^/b2—  1=  2 c2  + 2 ci^c2  — 1)^  — 1 = (<?+  Vc2—  l)2, 
and  the  capacity  in  farads  per  thousand  feet  of  conductor  is 


c _8.47iT10-9 

2 cosh-1 c 

Pender  and  Osborne  have  shown  that  for  parallel  cylindrical 
conductors  removed  from  the  influence  of  other  conductors,  the 


* McMahon,  Hyperbolic  Functions,  Art.  19. 


930 


ALTERNATING  CURRENTS 


distribution  of  the  charge  around  the  cross-section  which  is 
required  to  keep  the  surfaces  equipotential  surfaces,  gives  a 
surface  density  of  the  charge  on  each  conductor  which  conforms 
to  an  ellipse  with  its  major  axis  in  the  line  joining  the  centers 
of  the  two  conductors.* 

The  capacity  with  respect  to  ground  of  an  overhead  conduc- 
tor may  be  deduced  from  the  reasoning  applied  to  two  parallel 
conductors.  Suppose  the  conductor  of  diameter  2 a is  erected 
at  a distance  h above  the  surface  of  the  earth  and  substantial!}' 
parallel  therewith,  the  distance  h being  many  times  larger  than 
a.  The  capacity  of  1000  feet  of  this  conductor,  with  respect 
to  a hypothetical  equal  conductor  or  image  at  a distance  h 
below  the  surface  of  the  earth,  would  be 


C = 


3.68  xlO-9 


lo 


2 h ’ 


olO 


in  which  K becomes  unity  because  the  dielectric  is  air.  But 
the  surface  of  the  earth  corresponds  with  the  plane  of  zero 
potential  which  is  halfway  between  two  conductors,  and  the 
difference  of  potential  between  the  conductor  concerned  and 
the  earth  with  a given  charge  Q on  the  conductor  is  one  half 
as  great  as  the  difference  of  potential  between  the  couductor 
and  its  image.  Therefore  the  capacity  of  the  conductor  with 
respect  to  earth  in  farads  per  thousand  feet  of  length  is  twice 
the  foregoing,  or 

7.35  x IQ-9 


C = 


, 2 h 

logio— " 
a 


The  foregoing  expressions  for  the  capacity  of  cylindrical 
conductors  in  concentric  and  excentric.  relations  are  directly 
applicable  to  the  conditions  of  underground  and  overhead  wires. 
The  formulas  applying  to  cylinders  one  inside  the  other  apply 
not  only  to  concentric  cables,  but  are  also  useful  for  computing 
the  capacity  between  the  conductor  or  conductors  of  a lead- 
sheathed  underground  cable  and  the  sheath.  The  formulas 
are  developed  for  the  relations  of  two  conductors  with  respect 

* Pender  and  Osborne,  The  Electrostatic  Capacity  between  Equal  Parallel 
Wires,  Electrical  World , 1010,  Yol.  50,  p.  067. 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


931 


to  each  other,  but  the  same  reasoning  may  be  extended  to  mul- 
tiple conductor  cables.* 

The  capacity  current  set  up  by  each  harmonic  of  voltage  im- 
pressed between  the  conductors  is 

I = 2 irfCEl, 


where  E is  the  effective  value  of  the  harmonic  and  l is  the 
length  of  the  wire  in  thousands  of  feet.  The  total  charging 
current  is  the  square  root  of  the  sum  of  the  squares  of  all  the 
harmonic  currents.  The  minimum  charging  current  therefore 
occurs  when  the  impressed  voltage  is  sinusoidal. 

When  dealing  with  polyphase  circuits,  the  capacity  between 
conductors  is  computed  by  the  foregoing  formulas,  just  as  for 
single-phase  circuits ; but  the  disturbing  effect  on  the  electro- 
static field  of  the  additional  conductors  and  the  relation  of  the 
mesh  current  to  the  star  current  must  be  borne  in  mind  when 
computing  the  charging  current  flowing  in  a conductor.  Con- 
sidering a three-phase  circuit  with  the  conductors  at  the  corners 
of  an  equilateral  triangle  and  balanced  voltage,  the  electrostatic 
capacity  of  one  of  the  conductors  with  respect  to  the  other  two 
is  four  thirds  the  capacity  with  respect  to  only  one  of  the  others  ; 
but  the  voltage  impressed  on  this  capacity  is  the  voltage  re- 
action from  the  first  conductor  to  the  average  or  mid-voltage 


of  the  other  two,  which  is  equal  to  -~-E  if  E is  the  line  voltage. 


o 


Therefore,  in  a balanced  three-phase  circuit  with  equal  capacity 
for  each  pair  of  conductors,  the  charging  current  flowing  in  a 


2 

conductor  is  equal  to  -^-(2  TrfCEl^),  when  C is  the  capacity 
~\/  3 


per  pair  of  conductors,  E is  the  delta  line  voltage,  and  is  the 
length  in  thousands  of  feet  of  each  conductor  of  the  three-phase 
circuit.  This  gives  the  same  charging  circuit  per  conductor  of 
the  three-phase  circuit  as  would  be  obtained  by  connecting,  from 
each  conductor  to  the  neutral  point,  a condenser  of  capacity  2 C 
microfarads  per  1000  feet  of  the  line  (not  per  1000  feet  of 
wire),  in  which  case  the  voltage  impressed  on  each  condenser 

would  be.  -^=, 

V3 


* Russell,  Alternating  Currents,  Yol.  I.  Heaviside,  Electrical  Papers , Vol.  I, 
p.  42. 


932 


ALTERNATING  CURRENTS 


216.  Distributed  Resistance,  Self-inductance,  Leakage  Conduct- 
ance, and  Electrostatic  Capacity.  — The  relations  of  resistance, 
self-incluctance,  and  capacity  which  have  heretofore  been  con- 
sidered in  this  book  have  associated  the  quantities  as  though 
they  could  be  segregated  in  a circuit,  and  this  is  a sufficiently 
accurate  hypothesis  for  most  conditions  of  low  voltage  com- 
mercial circuits.  In  long  distance  transmission  lines  of  very 
high  voltage,  and  in  some  high  voltage  machines,  however,  the 
distributed  quantities  bring  in  additional  effects  which  will 
be  briefly  discussed  here.  The  complete  discussion  of  these 
phenomena  belongs  more  particularly  to  the  special  field  of 
power  transmission,  and  not  within  the  scope  of  this  work,  and 
the  following  discussion  therefore  only  gives  the  fundamental 
relations. 

In  the  case  of  two  parallel  uniform  conductors,  it  is  obvious 
that  each  element  of  length  of  each  conductor  possesses  a certain 
self-inductance,  and  likewise  that  each  element  of  length  of 
each  conductor  possesses  a certain  electrostatic  capacity 
measured  between  the  conductors.  These  elements  of  self- 
inductance and  capacity  being  uniformly  distributed  in  the 
case  of  the  uniform  conductors,  the  distributed  self-inductance 
and  capacity  bear  relations  to  each  other  in  transmitting  power 
which  are  analogous  to  the  mass  and  compressibility  of  a 
vibrating  uniform  metal  rod  or  string. 

The  drop  of  voltage  along  an  element  of  length  of  a con- 
ductor when  a variable  current  flows  is 

de  = Ridx  + Ldx—, 
dt 


where  R and  L are  the  resistance  in  ohms  and  self-inductance 
in  henrys  of  the  circuit  per  unit  (such  as  a centimeter,  a thou- 
sand feet,  or  a mile)  of  length  of  conductor  and  dx  is  an  element 
of  the  length.  Therefore  with  equal  uniform  conductors  com- 
posing the  circuit,  the  rate  of  change  of  voltage  between  the 
conductors  as  a function  of  distance  along  the  circuit,  that  is,  the 
space-rate  of  change  of  voltage  between  the  two  conductors,  is 


de 

dx 


= Ri  + L- 


di 

dt 


(1) 


Also,  the  loss  of  current  caused  by  the  uniformly  distributed 
capacity,  and  by  leakage  from  conductor  to  conductor  through 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


933 


the  insulation  and  by  convection  or  corona  effects,  likewise 
assumed  to  be  uniform  along  the  circuit,  is  for  each  element 

di  ==  Gredx  -f  C dx—, 
dt 

where  G-  and  0 are  the  conductance  in  mhos  of  the  insulation* 
and  the  capacity  in  farads  per  unit  of  length  of  conductor. 
Therefore,  the  space-rate  of  change  of  current  with  distance 
along  a conductor  is 

— =#<;  + <7—.  (2) 

dx  dt 

Differentiating  equations  (1)  and  (2)  with  respect  to  time 
and  then  with  respect  to  distance  gives  the  following  four 
differential  equations  of  the  second  order: 

d2e  _ di  Pi 

dx  • dt  dt  dt2' 

d2i  _g.de  g d2e 

dx  • dt  dt  dt2' 

d2e  _ di  g Pi 

dx2  dx  dx  • dt' 

(Pi  _ de  g d2e 

dx2  dx  dx  • dt 


By  eliminating 


(Pi 


dx  ■ dt 


from  the  second  and  third  of  these 
di 


and  substituting  the  value  of  ~ given  in  equation  (2),  there 
, , dx 

results 


LC~  + (RC+  LG~)—  + RGe  = ,:pe 

dt 2 J dt 


(3) 


A similar  process  of  elimination  of 


d2e 


dx2 

from  the  first  and 


dx  • dt 

fourth  of  the  group  of  four  equations  and  substituting  the  value 

of  — from  equation  (1)  gives 
dx 


LC^l  + (RC+LG)c^+  RGi  = '^-- 

dt 2 dt  dx 1 


(4) 


If  the  impressed  voltage  is  assumed  to  be  sinusoidal  at  the 
generator  with  a frequency  corresponding  to  2 irf  — <a,  equa- 


* The  insulation  conductance  in  mhos  is  the  reciprocal  of  the  insulation  re- 
sistance measured  in  ohms. 


934 


ALTERNATING  CURRENTS 


tion  (3)  may  be  solved  in  the  following  form,  when  neglecting 
the  transient  state, 

ex  — M \e~ax  f A sin  (a )t  — bx  + 7)  \ 

_l_  e-a(i-x ) ^ ft  s[n  y)  | . (5) 

This  shows  that  the  voltage  measured  between  conductors 
at  any  point  on  the  line  is  equal  to  the  sum  of  a wave  de- 
creasing logarithmically  as  the  length  x of  conductors  between 
the  generator  and  point  is  increased  and  a wave  decreasing 
logarithmically  as  the  length  l — x of  conductors  from  the 
receiver  at  the  far  end  of  the  line  is  increased.  The  former 
wave  may  be  looked  upon  as  a disturbance  or  wave  sent  out 
from  the  generator  and  the  latter  as  the  return  wave  produced 
by  reflection  of  the  original  wave  at  the  far  end  of  the  line. 
In  this  formula  l represents  the  total  length  of  conductor  in 
the  chosen  units  and  M,  A , B,  a,  b , and  7 are  quantities  deter- 
mined by  the  resistance,  self-inductance,  leakage  conductance, 
and  electrostatic  capacity  of  the  line  and  the  frequency  of  the 
impressed  alternating  voltage.  If  the  circuit  is  of  infinite 
length,  the  second  term  of  the  right-hand  member  of  equation 
(5)  disappears  for  all  finite  values  of  x. 

If  all  of  the  resistance,  self-inductance,  leakage  conductance, 
and  capacity  in  the  circuit  have  constant  values  which  are  in- 
dependent of  the  current  or  voltage,  the  generator  current  after 
the  expiry  of  the  transient  state  must  as  a consequence  of  the 
symmetry  of  the  equations  be  sinusoidal  when  the  generator 
voltage  is  sinusoidal,  but  the  trigonometrical  terms  in  the  solu- 
tion for  current  corresponding  to  equation  (5)  contain  an  addi- 
tional term  representing  the  angle  of  lag  of  the  current. 

Equation  (5)  shows  that  the  voltage  at  any  point  is  a sine 
function  with  respect  to  time,  since  at  a fixed  point,  x is  con- 
stant and  t is  the  only  variable.  The  period  of  this  function 

2 7T  1 

is  - — = - seconds,  which  is  the  period  of  the  generator  voltage. 

® / 

On  the  other  hand,  the  equation  shows  that  the  voltage  at  a 
given  instant  of  time  varies  along  the  line  as  a sine  function  of 
x multiplied  by  a logarithmic  decrement,  and  that  the  wave 
length  in  the  units  of  length  in  which  x is  measured  is 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


935 


That  is,  through  the  range  of  values  of  x extending  from  zero 
units  of  conductor  length  (kilometers,  miles,  or  otherwise  as  the 
2 77* 

case  may  be)  to  units  of  length,  the  voltage  goes  through 
o 

a complete  cycle  of  values  which  is  of  the  form  of  a sinusoid 
multiplied  by  a logarithmic  factor.  The  cycle  is  repeated  along 

O 

each  additional  units  in  length  of  conductor. 

b 

Since  the  voltage  and  current  at  any  fixed  point  go  through 
sinusoidal  cycles  with  a frequency  cycles  per  second,  and 

1 7 T 

2 77* 

points  distant  - — units  of  length  from  each  other  go  through 

b 

their  phases  simultaneously,  it  is  manifest  that  the  voltage  and 
current  cycles  are  in  the  nature  of  disturbances  which  travel 

2 r 

along  the  conductors  at  a rate  which  traverses  — — units  of  dis- 

b 

2 77* 

tance  in  - — seconds.  Hence,  the  velocity  of  propagation  of 

CO 

the  wave  in  distance  traveled  per  second  is 


CO  „ 

v = 7i=f\. 

The  solution  of  equations  (3)  and  (4)  which  are  most  con- 
venient for  purposes  of  computation  are  obtained  in  terms  of 
the  functions  of  hyperbolic  trigonometry,  as  tvas  early  pointed 
out  by  Kennedy  and  McMahon.*  When  the  voltage  and  current 
at  any  point  on  the  line  are  sinusoidal  functions  of  time,  the 
vector  relations  of  e and  i with  respect  to  their  derivatives  for 
a given  value  of  t may  be  written 

e = em  sin  cot , 

— • = ema ) cos  cot  =jcoe , (6) 

dt 

§ = - sin  tot  = — co2e  ; (7) 

dt 1 

and  correspondingly  for  the  current,. 

i = im  sin  ( cot  — 6 ), 

* McMahon,  Hyperbolic  Functions,  Art.  37.  Pender  has  worked  out  comput- 
ing formulas,  using  the  solution  with  circular  trigonometry.  See  Electrical 
World,  Yol.  56,  p.  667. 


936 


ALTERNATING  CURRENTS 


di  . . 
HI  =Jm-> 


(6') 


(7') 


Substituting  these  values  of  the  derivatives  in  equations 
(3)  and  (4)  gives  a vector  equation  which  is  independent  of  t 
and  in  which  the  values  of  e and  i may  therefore  be  taken  as 
either  maximum  or  effective  vector  values, 


and 


= ( - co2L 0 + >(P  C + L G)  + P G) E 
= ( P G A-  j<oC) E ; 

=(«+y»£>(ff+>c)i 


Writing  P for  (E  +ja>Ly(Gr  +jcoC)2  in  the  last  two  equa- 
tions gives 

d2E 
dx2 

Pi 

dx2 


and 


= P2E 
= P2I. 


The  solutions  of  these  equations  are 

E = A cosh  Px  -f-  B sinh  Px,  (8) 

and  1—  A!  cosh  Px  + B'  sinh  Px.*  (9) 

To  find  the  constants,  put  x = 0,  when  E and  I become  the 
generator  voltage  E0  and  generator  current  I0.  Therefore, 
since  sinh  Pa*  =0  and  cosh  Px  = 1 when  x = 0,  A = E0  and 
A!  = I0.  Also,  according  to  equations  (1)  and  (6'), 

= — (B  +j(i>L)I=  — ( B +jwL')QA'  cosh  Px  + B'  sinh  Px), 
dx 

the  negative  sign  being  required  here  because  of  the  vector 
relations,  (P  + ja>L)I  being  a counter  voltage  measured  in 
the  opposite  direction  from  x;  and  by  differentiation  of  equa- 
tion (8), 

— = AP  sinh  Px  + BP  cosh  Px. 
dx 


Hence,  AP  sinh  Px  = — (P  +ja>L)B'  sinh  Px , 

and  BP  cosh  Px  = — (P  4 -jcoL)A'  cosh  Px. 

* McMahon,  Hyperbolic  Functions,  Art.  14. 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


937 


Therefore 

B _ _ R +jwL  j , _ _ VR  +,jcoL  A,  _ 

P ^/G+jcoC 

P A 7...:  vg+j^ca  = _ 

R+jmL  WR+jwL 


■ZtA'=-ZX, 


and  B'  = — 


A 

Zi 


% 

zt 


Zt  being  put  for  the  impedance  of  the  line  per  mile  of  con- 
ductor, (R  +jo)L')^(  Cr  +jco  C)~'K 
Consequently,  equations  (8)  and  (9)  become 


E = E0  cosh  Px  — ZJq  sinh  Px , 

and 

1=  P cosh  Px  — sinh  Px. 
Zi 


If  a;  is  measured  from  the  receiver,  when  x = 0,  A = Et  and 
A’  = in  which  Et  and  Ix  are  the  voltage  and  current  at  the 

dE  — 

receiver.  Also  in  this  case  — = + ( R +jcoL)I , and  therefore 

ax 


- - E, 

B=  ZtIt  and  Br  = —■  Hence  the  equations  representing  the 
Zi 

voltage  and  current  at  any  distance  x from  the  receiver,  in 
terms  of  the  voltage  and  current  at  the  receiver,  are  like 
the  foregoing  except  that  Et  and  Ix  are  substituted  for  E0  and 
I0  and  the  negative  sign  before  the  second  term  of  the  right- 
hand  member  changes  to  a positive  sign. 

Since 


P = (R  + j<*L)  %(Gr  +jcoC)%=  \RGr  — co2LC  +j(i)(RC  + LGr)\ *, 

the  foregoing  equations  contain  hyperbolic  functions  of  com- 
plex arguments,  which  it  is  desirable  to  convert  into  functions 
of  real  arguments.*  Putting 

\RGr  — coPLC+jcoQRC  + X£r)|*  =a  + j/3, 

gives 

E = E0  cosh  (u  +j/3)x  — ZJ0  sinh  (a  + jf3~)x, 

— — jE 

1 = /0  cosh  (a  -f-  jfi)x  — =5  sinh  (a  +jf, 3)cc. 

Zi 


* This  is  for  the  purpose  of  convenience  in  computation  with  the  mathemati- 
cal tables  that  are  ordinarily  available,  but  tables  are  being  developed  by 
Kennedy  and  others  which  may  make  it  more  practicable  to  compute  directly 
from  the  hyperbolic  functions  with  complex  arguments. 


938 


ALTERNATING  CURRENTS 


Since 

sinh  (ax  + jfix)  = sinh  ax  cos  fix  +j  cosh  ax  sin  fix* 

and 

cosh  ( ax  + j fix')  = cosh  ax  cos  fix -\-j  sinh  ax  sin  fix , 
the  foregoing  convert  to 
E=  E0  cosh  ax  cos  fix  — ZlL0  sinh  ax  cos  fix 

+ j(E0  sinh  ax  sin  fix  — Zjl^  cosh  ax  sin  fix ),  (10) 


— — E 

I — E cosh  ax  cos  fix  — ~ sinh  ax  cos  fix 
° Zt 


- _£> 

+j(I0  sinh  ax  sin  fix  — ■=?  cosh  ax  sin  fix').  (11) 
Zi 


These  may  be  computed  by  the  ordinary  tables  of  functions 
of  circular  and  hyperbolic  trigonometry,  such  as  the  Smithsonian 
Mathematical  Tables.  A twenty-inch  slide  rule  and  a set  of 
tables  make  computation  by  this  formula  relatively  easy. 

The  values  of  « and  of  fi  may  be  computed  from  the  original 
expression  which  it  has  been  convenient  to  represent  by  a +jfi, 
that  is, 

f BG-<02LC+j<o(RC+LG-)\1=a+  jfi , 
from  which  comes 


R Gr  — <A?L  C + ja>(^R  C+  L Cr)  = («  +^/3)2, 

R a - gAL C + jco(RC  + LG)  = a2  — fi2  + 2 jafi, 
and  a2-  fi2  = RG  - co2LC, 

2 jafi  =jco(RC+  LG). 


Whence,  taking  positive  and  real  values  only,  as  being  appli- 
cable to  the  premises  here  under  consideration, 

a = — [ V(  U2  + co2L2  )(G2  + co2  C2)  + (RG  - a>2LC)f 
V2 

and 

fi  = -4;  r ^(R2  + o>2L2)(G2  + a>2C2)  - (RG  - O >2Z<7)] l- 
V2 

Of  these  terms,  a is  the  coefficient  a associated  with  x in  the 
exponential  terms  of  equation  (5).  It  is  called  the  Attenuation 
constant,  since  it  determines  the  rate  of  decrement  with  distance 
due  to  the  exponential  terms  of  the  wave  formula.  The  term 
* McMahon,  Hyperbolic  Functions,  Arts.  28-30. 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


939 


/3  is  the  coefficient  b associated  with  x in  the  sine  terms  of 
equation  (5),  and  is  called  by  Fleming  the  Wave  length  constant 
since  the  length  of  the  space  wave  of  voltage  or  current  is 
2 77- 

equal  to  -g-  • The  unit  in  which  the  wave  length  is  measured 

depends  on  the  unit  of  length  corresponding  to  R , L , Cr,  and 
C.  If  these  are  measured  in  ohms,  henrys,  mhos,  and  farads 
per  mile  of  conductor,  the  last  two  being  measured  from  con- 
ductor to  neutral  plane,  then  the  space  wave  length  is 

A.  = — miles  of  conductor. 

/3 


The  velocity  of  propagation  of  the  space  wave  is  then 


v = ^ = /\  miles  of  conductor  per  second. 

The  quantity  P = a +j  (3  has  been  called  the  Propagation  con- 
stant of  a circuit,  since  it  is  a complex  quantity  made  up  of  the 
attenuation  constant  and  the  wave  length  constant. 

The  imaginary  roots  of  the  equations  on  the  preceding  page 
lead  to  solutions  representing  oscillatory  phenomena. 

To  obtain  the  voltage  and  current  at  the  receiving  end  of  a 
transmission  line  when  the  current  and  voltage  at  the  sending 
end  are  known,  it  is  only  necessary  to  substitute  the  total  length 
of  a line  conductor  in  the  place  of  x in  equations  (10)  and  (11), 
giving, 

E1  — R0  cosh  al  cos  (31  — ZtI^  sinli  al  cos  /3Z 
+j (P0  sinh  al  sin  /3l  — ZtI0  cosh  al  sin  /3l), 


Ix  = ^ = I(t  cosh  al  cos  /3l  — ^ sinh  al  cos  (31 


+j[  I0  sinh  al  sin  /3l  — n?  sinli  al  cos  (31  ), 


(12) 


(13) 


Ev  Iv  and  Zx  being  the  voltage,  current,  and  impedance  at  the 
receiver. 

Eliminating  I0  from  (12)  and  (13)  gives 


Z(  sinh  al  cos  (31  + Z)cosh  al  cos  f3l  + j{Zt  cosh  al  sin  j3l  -1-  Zx  sinh  al  sin  (31 ) 


940 


ALTERNATING  CURRENTS 


and  the  ratio  of  /x  to  J0  is 

= ^ 

I Z[  cosh  cd  cos  (il  + Zx  sinh  al cos [il  + j(Zl  sinh  al  sin  (il  + Zx cosh  al  sin  /il)  ’ 

(15) 

from  which  the  generator  voltage  and  current  may  be  computed 
if  the  characteristics  of  the  line  and  the  voltage,  current,  and 
impedance  of  the  receiver  are  given. 

...  — !<} 

It  is  to  be  observed  that  Zt  is  quite  different  from  Z0  = P& . 

I, 

The  former  depends  only  on  the  frequency  of  the  current  and 
the  electrical  constants  of  the  line  conductors  per  unit  length. 
It  is  independent  of  the  length  of  the  line  and  of  the  character 
of  receiver  into  which  the  line  delivers  power.  The  impedance 
Z0,  on  the  other  hand,  depends  on  the  length  of  the  line  and 
the  character  of  the  receiver  in  addition  to  being  dependent  on 
the  constants  of  the  line  per  unit  length.  For  a line  of  infinite 
length  Zt  = Z0,  but  for  an  ordinary  line  they  bear  a complex 
relation  to  each  other.  A relation  that  is  sometimes  useful  to 
know  is  the  ratio  of  the  generator  current  when  a short  circuit 
occurs  at  the  receiver  to  the  generator  current  in  normal  work- 
ing. The  ratio  is 

T{)  _ tanh  (PI  + 6) 

T tanh  PI 


in  which  P 0 and  I0  are  the  generator  current  respectively  with 
the  receiver  end  of  the  line  short-circuited  and  in  normal  work- 
ing, and  6 = tanh-1^1. 

The  effect  of  either  a short  circuit  or  an  open  circuit  at  anjr 
point  is  readily  observed  from  these  equations.  In  the  case  of 
a short  circuit  occurring  with  a length  x of  conductor  between 
it  and  the  generator,  P becomes  zero  at  that  point,  and 


E()  cosh  ax  cos  fix  — Z,Z0  sinh  ax  cos  fix 

+j(P0  sinh  ax  sin  fix—  Zj^  cosh  ax  sin  fix')  = 0. 


Therefore 


h = 


Z^fcosh  ax  cos  fix  +j  sinh  ax  sin  fix) 

. t 9 

Z/sinh  ax  cos  fix  +j  cosh  ax  sin  fix) 


and 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


941 


I — ZJ0  

-a~x  — - — . — 

.Zi  cosh  az  cos /ix  + Zjsinhaxcos  /to+A.Z'jsinhccrsin  fix  + ^cosh  to;  sin  fix) 

K 


Zj(sinh  ax  cos  / 3x  +j  cosh  ax  sin  fix') 

In  case  of  an  open  circuit  at  distance  x from  the  generator 
/becomes  zero  at  that  point,  and 

_ 

P cosh  ax  cos  fix  — sinh  ax  cos  fix 
Zi 

+j(l0  sinh  ax  sin  fix—  cosh  ax  sinh  fix\  = 0. 

^ Zl  J 


Therefore 


f _ A/n(sinh  ax  cos  fix  +j  cosh  ax  sin  fix') 

A)-  , _ ; r , . _ ’ 


and 


Er  = 


Z;(cosh  ax  cos  fix  +j  sinh  ax  sin  fix) 

K 

(cosh  ax  cos  fix  +j  sinh  ax  sin  fix) 


In  an  infinitely  long  cable  Ix  = / (cosh  Px  — sinh  Px)  = I0e~Pxi 
of  which  the  real  part  is  I(f~ax  and  therefore  Ix  = I0e~ax. 

As  pointed  out  in  an  earlier  paragraph,  the  space  wave  length 

2 7 r 

measured  in  conductor  miles  is  when  the  resistance,  in- 
ductance, insulation  conductance,  and  capacity  are  given  in 
terms  of  ohms,  henrys,  mhos,  and  farads  per  mile  of  conductor. 
For  conductors  of  the  ordinary  sizes  and  spacings  used  in  the 
long  distance  transmission  of  large  amounts  of  power,  and  cur- 
rents of  the  ordinary  frequencies  used  for  that  purpose,  a wave 
length  is  many  hundreds  of  miles  when  measured  along  the 
line.  For  instance,  an  ungrounded  3-phase  overhead  line  of 
No.  0 copper  wires  spaced  ten  feet  apart  has  resistance  of  ap- 
proximately .52  ohm  per  mile  of  conductor,  self-inductance 
of  approximately  .0022  henry  per  mile  of  conductor,  leakage 
conductance  negligible  at  ordinary  altitudes  if  the  line  voltage 
is  below  125,000  volts,  and  capacity  of  .0078  xlO-6  farads  per 
mile  of  conductor.  This  gives  for  the  value  of  the  wave  length 
constant  when/=  25  cycles  per  second  or  co  = 157  radians  per 
second,  fi  = .00077 

and  for  the  wave  length 

2 7r 

A = = 8150  miles  of  conductor. 

fi 


942 


ALTERNATING  CURRENTS 


The  wave  length  in  miles  of  line  is  one  half  as  great  as  the 
length  in  miles  of  conductor,  since  the  constants  are  all  given  on 
the  assumption  of  an  outgoingconductor  and  return.  No  over- 
head transmission  line  has  )ret  approached  this  length,  or  even 
the  length  of  1925  miles,  which  are  the  miles  of  line  required 
to  give  a complete  wave  length  if  the  frequency  were  60 
cycles  per  second  on  the  foregoing  line. 

Consequently,  it  is  possible  to  neglect  the  formulas  which 
recognize  the  effects  of  distributed  resistance,  self-inductance, 
leakage,  and  capacity  when  computing  long  distance  power 
transmission  lines  of  lengths  within  the  limits  of  existing  prac- 
tice, and  to  make  the  computations  on  the  assumption  that  the 
fall  of  potential  is  uniform  along  the  conductors.  Leakage 
being  usually  negligible,  the  effects  of  the  electrostatic  capacity 
may  be  approximated  by  assuming  the  capacity  of  the  conduc- 
tors concentrated  at  certain  points  instead  of  distributed.  For 
short  lines  it  is  sufficient  to  consider  the  capacity  all  concen- 
trated at  the  middle  of  the  line ; but  for  lines  having  lengths 
approaching  the  limits  of  the  existing  longest  lines,  it  is  some- 
times essential  to  make  a closer  approximation  to  truth,  and  the 
capacity  may  be  assumed  to  be  concentrated  in  three  divisions, 
one  sixth  of  the  whole  capacity  at  each  end  of  the  line  and  the 
remaining  two  thirds  at  the  middle  of  the  line. 

The  transmission  of  large  amounts  of  power  over  under- 
ground cables  has  not  yet  reached  lengths  of  circuit  approach- 
ing the  lengths  in  individual  overhead  lines,  and  the  voltage 
impressed  on  cables  is  usually  less  than  20,000  volts.  The  self- 
inductance of  the  conductors  in  the  usual  underground  circuit 
is  much  less  than  the  self-inductance  of  a corresponding  over- 
head circuit,  on  account  of  the  closer  spacing  of  the  conductors ; 
but  the  electrostatic  capacity  of  the  cabled  conductors  is  much 
greater  than  when  the  circuit  is  overhead,  on  account  of  the 
closer  spacing  of  the  conductors  and  the  relatively  high 
dielectric  constant  of  the  solid  insulating  material.  An  un- 
grounded 3-phase  cable  of  three  symmetrically  spaced  No.  0 
wires  with  rubber  insulation  having  ^ inch  thickness  of  wall 
around  each  conductor  has  a resistance  of  approximately  .52 
ohm  per  mile  of  conductor,  a self-inductance  of  approximately 
.00065  henry  per  mile  of  conductor,  leakage  negligible,  and 
a capacity  of  approximately  .0545  x 10  ~6  farads  per  mile  of 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


943 


conductor.  The  wave  length  constant  when  the  frequency  is 
twenty-five  cycles  per  second  is  therefore  /3  = .001645  and  the 
wave  length  in  miles  of  conductor  is 


This  corresponds  with  1910  miles  in  length  of  cable.  This  far 
exceeds  the  lengths  of  circuit  ordinarily  found  in  underground 
cables  for  heavy  electric  power  transmission,  hut  in  the  case  of 
the  longer  underground  cables  used  with  voltages  higher  than 
a few  thousand  volts,  it  is  desirable  to  utilize  the  exact  formu- 
las for  determining  the  effects  of  capacity. 

In  the  case  of  long  distance  telephone  service  the  situation 
is  different  on  account  of  the  higher  frequency  of  the  current 
harmonics  which  it  is  desirable  to  retain  without  great  attenua- 
tion, and  also  because  it  is  desirable  to  adjust  the  relation  of 
resistance,  self-inductance,  and  capacity  so  that  the  attenuation 
and  the  velocity  of  propagation  shall  be  substantially  the  same 
for  current  harmonics  of  a considerable  range  of  frequencies 
(such  as  from  200  periods  per  second  to  1800  periods  per 
second),  in  order  that  the  intelligibility  of  the  irregular 
speech  waves  may  be  maintained.  This  requires  what  is 
called  a Distortionless  circuit  and  the  exact  formulas  are  needed 
in  the  computation  of  such  circuits. 

When  leakage  is  negligible,  the  values  of  a and  ft  heretofore 
given  reduce  to 


If  the  inductive  reactance  a>L  is  at  the  same  time  negligibly 
small  compared  with  the  resistance,  these  reduce  to 


approximately,  since  the  square  root  of  a -f  b is  very  nearly 


A = — = 3820. 

/3 


944 


ALTERNATING  CURRENTS 


equal  to  Va -| — when  a is  large  compared  with  b , and  when 

2V  a 


leakage  is  negligible  under  these  circumstances, 


Also,  leakage  being  negligible,  if  2 wL  is  very  large  compared 
with  R2, 

0 = coVCL, 

1 


and 


« _ 


0 V LC 


Whence,  by  making  wL  for  the  various  harmonics  of  a periodic 
current  very  large  compared  with  R , the  attenuation  and  the 
velocity  of  wave  propagation  of  the  harmonics  are  both  made 
independent  of  the  frequency  of  the  harmonics  and  dependent 
solely  on  R , X,  and  C. 

The  latter  condition  may  also  be  accomplished  vrhen  leakage 
is  not  negligible,  provided  LG=  RC , except  that  attenuation 
and  velocity  of  propagation  are  dependent  upon  R , L , 6r,  and 
C instead  of  R , X,  and  (7,  only.  In  this  case,  the  original  ex- 
pressions for  a and  0 reduce  to 

«= -L  [V(ze<7  + co2Lcy  + a>\La  - Rcy+<iRa-«)2Lcy]\ 
V2 

= y/RG, 

/3  = -1  [ V(i26^  + <o2LC)2  + <d\LG-  RC)2  - (RG-a>2LCrf 

V2 

= coV LC. 

The  importance  of  this  sort  of  treatment  in  the  case  of  a 
long  distance  telephone  line  may  be  recognized  from  the  facts 
that  an  overhead  line  of  No.  8 B.  & S.  gauge  wire  spaced  twelve 
inches  apart  has  resistance  of  3.3  ohms  per  mile  of  conductor, 
self-inductance  of  2 x 10~3  henrys  per  mile  of  conductor,  capac- 
ity of  .004  x 10-6  farads  per  mile  of  conductor,  and  the  insu- 
lation resistance  ought  to  be  in  the  order  of  some  megohms  per 
mile  of  conductor;  and  that  it  is  desirable  to  maintain  the 
attenuation  and  velocity  of  propagation  substantially  the  same 
for  harmonics  of  the  voice  currents  within  the  range  of  200  to 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


945 


1800  cycles  per  second.  The  foregoing  constants  give  wave 
lengths  which  are  but  fractional  compared  with  the  length  of 
many  long  distance  telephone  circuits. 

In  the  case  of  telephone  cables,  the  relations  are  even  more 
impressive.  With  the  usual  lead-covered  cables  with  loose  dry 
paper  wrapping  for  the  conductors,  the  constants  for  No.  19 
B.  & S.  gauge  conductors  are  per  mile  of  conductor  approxi- 
mately 45  ohms,  5 x 10-3  henrys,  .035  x 10-6  farads,  and  insu- 
lation resistance  of  one  half  a megohm  or  more. 

This  gives  a wave  length  of  less  than  sixty  miles  of  cable  for 
the  harmonic  corresponding  to  an  angular  velocity,  &>,  of  5000 
radians  per  second  (/=  800  periods  per  second,  approximately), 
which  is  about  a mean  value  for  the  harmonics  of  voice  currents 
that  are  necessary  to  retain  an  approximately  unaltered  relative 
magnitude  during  transmission  in  order  that  the  transmitted 
speech  may  be  fully  articulated  by  the  receiver.  For  the  har- 
monic of  frequency  1800  periods  per  second,  the  wave  length 
on  this  cable  is  less  than  thirty-five  miles.  Since  telephony 
through  underground  cables  is  now  practiced  over  many  tens  of 
miles,  the  transmission  computations  must  obviously  be  executed 
with  the  formulas  which  include  all  the  features  of  the  wave 
transmission.* 

The  foregoing  formulas  relate  to  uniform  circuits.  They 
may  be  applied  to  each  uniform  part  of  a non-uniform  circuit, 
provided  the  voltage  impressed  on  the  part  concerned  and  the 
vector  impedance  of  the  succeeding  part  are  known;  but  changes 
of  character,  especially  if  they  are  abrupt,  cause  reflection  points 
which  greatly  alter  the  distribution  of  voltage  compared  with 
the  distribution  of  voltage  over  a uniform  section  of  the  same 
length. 

217.  Analogies  of  the  Electric  Circuit.  — The  deductions  of 
Chapter  VI  have  shown  very  clearly  that  concentrated  self-in- 
ductance and  concentrated  capacity  in  a circuit  may  be  made 
to  neutralize  each  other  when  a sinusoidal  voltage  is  applied  to 
the  circuit,  and  the  self-inductance  and  capacity  are  constant. 
In  this  case  the  self-inductance  and  capacity  act  in  opposition, 
so  that  at  each  instant  energy  is  being  restored  or  released  in 

* The  simpler  problems  .of  long  distance  telephone  transmission  are  dealt 
with  admirably  by  Fleming  in  a book  entitled  The  Propagation  of  Electric 
Currents  in  Telephone  and  Telegraph  Conductors. 

3 p 


94  G 


ALTERNATING  CURRENTS 


the  magnetic  field  at  exactly  the  same  rate  as  energy  is  being 
released  or  stored  in  the  charge  of  the  condenser.  The  self- 
inductance and  capacity  may  therefore  be  said  to  supply  each 
other’s  demands,  and  the  power  delivered  by  a generator  to  the 
circuit  may  be  wholly  utilized  in  doing  work  on  a non-reactive 
receiver  such  as  incandescent  lamps  and  in  heating  the  wires  of 
the  circuit.  The  vector  power  which  is  transferred  back  and 
forth  between  the  self-inductance  and  capacity  may  be  many 
times  as  great  as  that  given  to  the  circuit  by  the  generator, 
and  volt-amperes  at  the  terminals  of  the  self-inductance  and  of 
the  condenser  must  then  be  proportionally  greater  than  the 
volt-amperes  delivered  by  the  generator. 

This  condition  can  fully  exist  only  when  the  frequency  of 

the  impressed  voltage  produces  the  relation  2 irfL  = ^ or 

2 7rfC 

LO  = and  the  natural  period  of  the  circuit  is  2 tt^/LC .j 

From  the  condition  2 rrfL—  — - — it  is  seen  that  - = 2 7rVZ6r. 

2 irfC  f 

The  natural  period  of  discharge  of  the  circuit  is  therefore  equal 
to  the  period  of  the  cycles  of  impressed  voltage,  or,  as  may  be 
said,  equal  to  the  rate  of  the  electrical  vibrations  impressed  on 
the  circuit  by  the  generator.  This  relation  between  the  vibra- 
tions of  the  line  and  of  the  generator  is  similar  to  that  of  a 
vibrating  tuning  fork  or  string  and  a sounding  board  when  the}* 
are  in  resonance,  and  therefore  the  term  Electrical  resonance 
lias,  on  account  of  the  analogy,  been  applied  to  the  electric 
circuit.  An  electric  circuit  is  said  to  be  in  resonance  with  an  im- 
pressed voltage  when  the  natural  period  of  the  circuit  is  equal  to 
the  period  of  the  impressed  voltage.  When  this  condition  exists, 
the  maximum  current  is  caused  to  flow  in  the  circuit  by  the 
application  of  a given  impressed  voltage,  the  value  of  the  cur- 
rent in  a resonant  circuit  from  which  no  external  work  is  sup- 
plied being 


If  the  self-inductance,  capacity,  and  resistance  are  in  series  in 
the  circuit,  it  is  evident  that  when  the  periods  of  the  circuit 


1 = 


U 


* Art.  69  (4) . 


t Art.  58,  Case  (2). 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


947 


and  the  impressed  frequency  are  in  resonance,  the  voltage  be- 
tween the  terminals  of  the  capacity  ^ = ^ is  a maximum, 

since  the  circuit  current  is  a maximum ; and  the  same  is  true  of 
the  voltage  between  the  terminals  of  the  inductance.  If  either 
the  frequency,  the  self-inductance,  or  the  capacity  is  changed  in 
value,  the  value  of  the  current  falls,  and  the  condenser  voltage 
falls,  unless  the  other  elements  are  changed  in  value  in  such  a 
way  as  to  continue  the  condition  of  resonance.  A condenser  in 
a resonant  circuit  may  be  used  as  a transformer  of  voltage  by 
connecting  non-reactive  apparatus  across  its  terminals,  as  has 
been  suggested  by  Blakesley,  Loppe  et  Bouquet,  Pupin,  and 
others. 

If  the  self-inductance  and  capacity  ai’e  in  parallel  in  the  cir- 
cuit, the  voltage  at  their  terminals  cannot  be  greater  than  that 
impressed  upon  the  circuit  minus  the  loss  of  voltage  in  the  lead 
wires;  but  when  the  circuit  is  resonant,  the  current  furnished 
to  the  circuit  by  the  generator  is  at  a minimum  which  is  equal 
to  the  power  consumed  by  the  circuit  divided  byT  the  impressed 
voltage,  while  the  current  transferred  between  the  inductance 
and  capacity  is  a maximum  which  may  be  a great  many  times 
as  great  as  the  value  of  the  generator  current. 

Resonant  circuits  in  the  hands  of  experimenters  such  as 
Hertz,  Lodge,  and  others  have  produced  remarkable  results, 
which  have  led  to  great  advances  in  our  knowledge  of  elec- 
tricity, while  mathematical  analysis  of  such  circuits  has  led  to 
further  discoveries.  These  results  have  caused  some  to  expect 
remarkable  effects  to  be  gained  from  the  use  of  resonant  cir- 
cuits (or  Tuned  circuits,  as  they  are  sometimes  called)  for  the 
purposes  of  the  electrical  transmission  of  power. 

Circuits  which  are  installed  for  the  transmission  of  power 
over  considerable  distances  (whether  the  wires  are  overhead  or 
underground)  always  contain  capacity  and  self-inductance  dis- 
tributed along  their  lengths.  It  would  be  possible  in  such  lines 
to  adjust  the  capacity  and  self-inductance  so  as  to  give  reso- 
nance, and  the  results  to  be  gained  from  so  doing  may  be  ex- 
amined through  analogy. 

A mechanical  analogue  of  an  electric  circuit  is  shown  in  Fig. 
523.  This  consists  of  a tube  fitted  with  two  plungers  and 
filled  with  a perfectly  elastic  fluid.  The  properties  of  this  fluid 


948 


ALTERNATING  CURRENTS 


may  be  used  to  represent  electrical  quantities  according  to  the 
analogies  ; fluid  velocity  — electric  current  ; fluid  pressure  — 
voltage  ; inertia  — self-inductance  ; compressibility  * — elec- 
trostatic capacity  ; frictional  resistance  — electrical  resistance. 
Now  suppose  the  fluid  to  be  without  inertia  and  perfectly  incom- 
pressible ; then  if  plunger  A is  moved  toward  _Z>,  a uniform  cur- 
rent is  instantly  set  up  in  the  whole  length  of  the  tube,  the  ve- 
locity of  which  is  equal  (in  proper  units)  to  the  pressure  applied 
to  the  plunger  divided  by  the  frictional  resistance.  If  plunger 
A is  caused  to  move  up  and  down  harmonically,  the  plunger  A' 


D a 


Fig.  523.  — Illustration  of  a Mechanical  Analogue  of  an  Electric  Circuit. 

at  the  other  end  of  the  line  will  have  an  exactly  equal  syn. 
clironous  harmonic  motion.  This  is  analogous  to  the  state  of 
an  electric  circuit  without  inductance  or  capacity.  Figure  524 
shows  diagrannnatically  the  state  of  the  circuit,  the  distance  of 
the  broken  line  from  the  heavy  line  being  equal  to  the  current 
at  each  point.  The  light  full  line  shows  the  gradual  fall  of 
pressure  between  A and  A',  caused  by  the  resistance,  and  the 
sudden  fall  of  pressure  at  Af,  caused  by  the  external  work  done 
by  plunger  A'. 

If  the  fluid  is  compressible  but  has  no  inertia,  it  is  evident  that 
the  motion  of  the  plunger  at  A!  will  be  less  than  that  at  A,  which 
is  analogous  to  the  decadence  of  current  as  it  flows  along  a circuit 
having  capacity,  due  to  the  quantity  of  electricity  entering  into 
the  electrostatic  charge  on  the  conducting  wires.  The  move- 
ments of  the  plungers  are  isochronous  but  not  in  synchronism. 

* Compressibility  of  a fluid  is  the  ratio  of  compression  (change  of  volume)  to 
the  pressure  producing  it,  and  electrical  capacity  is  the  ratio  of  the  charge 
(change  of  quantity)  to  the  voltage  producing  it. 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


949 


In  this  case  the  motion  of  the  plunger  A will  exert  its  maximum 
pressure  when  the  fluid  is  most  compressed,  or  at  the  end  of  its 
stroke  where  its  velocity  is  least.  Hence  the  velocity  of  the 
fluid  at  the  genera- 
tor (i.e.  the  genera- 
tor current),  which 
is  greatest  at  the 
middle  of  the 
stroke,  leads  the 


pressure  by  90°  of  Fig.  524.  — Illustration  showing  Conditions  of  Currents 
phase  The  mao'-  an<*  Pressures  when  the  Pistons  of  Fig.  523  move  in  an 
r " ” Incompressible  Fluid  which  is  without  Inertia. 

nitudes  of  the  cur- 
rent and  voltage  throughout  the  circuit  are  illustrated  in  Fig. 
525,  after  the  manner  explained  in  connection  with  Fig.  524. 

If  the  fluid  has  inertia  but  is  incompressible,  the  velocities  at 
A and  A'  are  equal ; that  is,  the  current  throughout  the  circuit 
is  uniform,  but  the  pressure  exerted  upon  piston  A is  greatest 
when  the  acceleration  is  greatest,  which  is  at  the  beginning  of 
the  stroke  where  the  velocity  is  least.  Consequently  the  cur- 
rent lags  behind  the  pressure  by  90°.  This  is  analogous  to  the 
electric  circuit  with  self-inductance  but  no  capacity. 

If  the  fluid  has  both  inertia  and  compressibility,  the  column 
of  fluid  in  the  tube  then  takes  upon  itself  the  usual  properties 

of  material  elastic 
bodies,  and  possesses 
a natural  rate  of  vi- 
bration of  its  own 
which  depends  upon 
the  dimensions  of 
the  column  and  the 


Fig.  525.  — Illustration  showing  the  Conditions  of  Cur-  inertia  atld  COmpreS- 
rents  and  Pressures  when  the  Pistons  of  Fig.  523  move  m-tj.  r j.i  a - i 
in  a Compressible  Fluid  having  No  Inertia.  Slblllty  °f  the  flulcb 

The  period  of  this 

vibration,  as  is  proved  in  elementary  mechanics,  is  proportional 
to  the  square  root  of  the  density  divided  by  the  elasticity,  or  to 
the  square  root  of  the  product  of  the  inertia  and  compressibility. 
Hence  T = a V MK  where  a is  a constant,  M mass,  K compressi- 
bility, and  T time  of  vibration. 

In  this  case,  if  the  plunger  A (Fig.  523)  is  moved  with  a 
sinusoidal  velocity  having  a period  of  T seconds,  which  is  the 


950 


ALTERNATING  CURRENTS 


same  as  the  natural  period  of  vibration  of  the  column  of  fluid, 
the  fluid  will  be  thrown  into  vibrations  which  require  one  com- 
plete traversal  of  the  circuit  to  make  a wave  length.  Hence, 
if  there  is  no  power  taken  from  the  circuit,  there  are  nodes  or 
points  of  no  motion  at  a and  a',  and  antinodes  or  points  of 
maximum  motion  at  the  plungers.  Since  the  directions  of  mo- 
tion in  the  two  halves  of  a wave  are  opposite,  the  two  plungers 
move  in  opposite  directions  in  the  tube.  As  the  velocity  of  the 
fluid  varies  from  node  to  antinode  as  a sinusoidal  function, 
the  loss  of  power  by  friction  is  reduced  to  one  half  the  value 
which  it  has  for  an  equal  plunger  velocity  in  the  inertialess, 
incompressible  fluid.  The  velocity  of  propagation  of  the  dis- 
turbance through  the  fluid  from  plunger  A to  A'  is  equal  to 

- centimeters  per  second,  where  l is  the  length  of  column. 

affMK 

Since  the  velocity  of  movement  of  the  fluid  falls  off  from  the 
plungers  toward  the  nodes,  the  pressure  upon  the  fluid  exerted 
by  the  plungers  must  be  proportionately  multiplied  at  the 
nodes,  in  order  that  the  same  power  may  be  transmitted  through 
the  fluid  across  the  nodes  as  is  applied  at  the  prime  plunger  A. 
The  condition  of  pressure  and  velocity  is  diagrammatically  rep- 
resented in  Fig.  526.  If  power  is  transferred  to  an  outside 


under  the  conditions  here  cited,  require  that  the  power  shall  he 
transferred  from  one  'plunger  to  the  other  tvholly  through  the  ab- 
sorption and  redelivery  of  the  power  by  the  fluid  by  means  of  the 
effects  of  inertia  and  elasticity.  The  fluid  must  therefore  have 
a sufficient  mass  so  that  even  at  the  slow  velocity  at  the  nodal 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


951 


points  its  kinetic  energy  shall  be  sufficient  to  carry  the  energy 
in  the  circuit  across  those  points. 

This  analogue  represents  the  conditions  in  the  resonant  elec- 
tric circuit  with  distributed  self-inductance  and  capacity. 
Carrying  in  mind  the  analogue  and  the  diagrammatic  represen- 
tation of  current  and  pressure  in  Fig.  526,  it  is  easy  to  draw 
definite  conclusions  in  regard  to  the  effect  of  resonance  on  the 
operation  of  electric  circuits  for  the  transmission  of  power. 

The  advantages  of  a resonant  circuit  for  electrical  transmis- 
sion are  then  : (1)  a gain  of  upwards  of  one  half  of  the  I2R  loss 
that  would  be  caused  by  the  transmission  of  an  equal  amount 
of  power  at  an  equal  receiving  voltage  over  the  same  circuit 
when  out  of  resonance  ; (2)  more  satisfactory  regulation  than 
would  be  found  in  a non-resonant  but  reactive  line,  since  the 
difference  in  voltage  between  generator  and  receiver  is  equal 
to  current  times  resistance  instead  of  current  times  an  imped- 
ance which  is  greater  than  the  resistance. 

The  principal  disadvantage  of  a resonant  circuit  for  electrical 
transmission  is : a very  large  excess  of  voltage  on  the  line  at 
certain  points,  or  nodes  of  current,  which  excess  decreases  to- 
ward the  antinodes.  If  satisfactory  resonance  is  to  be  gained  by 
adjusting  the  self-inductance  and  capacity  of  the  circuit  so  that 
the  voltage  at  the  nodes  of  current  is  no  greater  than  ten  times 
that  at  the  antinodes,  the  average  voltage  along  the  line  must  be 

caused  to  be  seven  times  that  of  the  antinodes,  using  a 

sinusoidal  function.  In  other  words,  if  the  voltage  which  is 
safe  for  use  is  limited  by  the  insulation,  we  may  say  that  the 
average  strength  of  insulation  on  the  line  must  be  seven  times 
as  great  as  would  be  necessary  at  the  generators.  This  enor- 
mous increase  of  insulation  must  be  made  to  save  fifty  per  cent 
of  the  I2R  loss  caused  by  the  transmission  of  a certain  amount 
of  power  over  a given  line.  A much  more  reasonable  plan  for 
heavy  power  transmission  would  be  to  reduce  the  self-inductance 
and  capacity  of  the  line  to  a minimum,  avoiding  resonance  and 
raising  the  generator  voltage  to  1.4  its  previous  value.  Now 
the  same  power  could  be  transferred  over  the  line  with  the 
same  resistance  as  before,  the  I2R  loss  being  the  same  as  when 
the  line  was  resonant,  but  the  average  strain  on  the  insulation 
would  be  only  one  fifth  as  great  as  in  the  resonant  line. 


952 


ALTERNATING  CURRENTS 


The  highest  voltage  which  can  be  economically  used  on  cir- 
cuits for  the  electrical  transmission  of  large  amounts  of  power 
over  long  distances  is  generally  conceded  to  be  set  at  the  limit 
which  may  be  properly  insulated.  If  this  is  true,  the  preced- 
ing paragraph  shows  that,  with  equal  insulation,  the  generator 

voltage  may  be  safely  made  — = times  greater  on  a non-resonant, 

V2 

long  distance  transmission  line  than  that  which  is  safe  on  a 
resonant  line,  where  X is  the  ratio  of  the  maximum  voltage  to 
the  generator  voltage  on  the  resonant  line.  This  shows  that 
the  non-resonant  line  would  be  by  far  the  most  economical  for 
long  distance  heavy  power  transmission,  even  if  it  were  com- 
mercially possible  to  maintain  resonance  on  service  circuits. 
For  the  distribution  of  power  over  short  distances,  the  voltage 
is  usually  quite  low,  and  the  insulation  limit  is  not  approached, 
so  that  resonance  might  be  introduced  without  adding  to  the 
insulation ; but  the  reactions  of  transformers  and  motors  on 
the  line  make  it  practically  impossible  to  keep  the  line  in  reso- 
nance. Similar  defects  are  seen  in  the  propositions  for  using 
resonant  lines  for  various  other  classes  of  electrical  transmission 
except  telephony. 

These  deductions  in  regard  to  resonance  have  been  made 
upon  the  assumption  of  sinusoidal  currents.  In  practice  these 
are  now  seldom  exactly  realized,  since  iron-cored  transformers 
and  motors,  and  tooth-cored  alternators,  introduce  distortions, 
and  a circuit  which  is  resonant  for  the  fundamental  wave  is  not 
resonant  for  its  harmonics.  As  the  question  of  resonance  now 
rests,  it  does  not  enter  into  problems  relating  to  ordinary  elec- 
tric power  circuits  in  such  a way  as  to  modify  practice  except 
as  it  in  certain  instances  causes  undue  stresses  on  the  insulation 
through  its  accidental  occurrence  in  connection  with  transient 
effects  caused  by  switching,  arcing  between  wires  or  from  wires 
to  ground,  effects  of  lightning  induction,  and  the  like. 

In  telephony  (which  consists  of  the  electric  transmission  of 
power  in  very  minute  quantities  under  certain  special  condi- 
tions), the  voltages  are  relatively  low  and  satisfactory  insulation 
for  resonant  transmission  may  be  readily  accomplished.  It  is, 
therefore,  good  practice  to  load  long  distance  telephone  lines 
with  self-inductance  by  inserting  coils  at  frequent  intervals,  for 
the  purpose  of  making  the  lines  more  nearly  distortionless. 


INDUCTANCE  AND  ELECTROSTATIC  CAPACITY 


953 


218.  Corona.  — One  of  the  earliest  observed  manifestations 
of  electricity  was  the  brush  discharge  between  the  electrodes  of 
an  electrostatic  machine  or  near  a highly  charged  pointed  body. 
It  was  early  recognized  that  this  is  accompanied  by  convection 
effects  whereby  a current  of  electrically  charged  air  is  caused 
to  flow  away  from  the  body  concerned,  with  a tendency  to  dis- 
charge the  body.  This  is  illustrated  by  holding  a lighted  candle 
near  a point  protruding  from  the  electrode  of  an  electrostatic 
machine  in  operation,  when  it  may  be  observed  that  the  candle 
flame  is  blown  aside  by  the  air  current  caused  by  the  particles 
of  air  receding  from  the  electrode  on  account  of  repulsion  after 
they  have  become  charged  by  contact.  The  phenomenon  is  ac- 
companied by  a silent  transfer  of  electricity  from  one  electrode 
of  the  machine  to  the  other  by  means  of  the  air  particles. 

While  the  usual  electric  circuits  of  commerce  were  only  of 
relatively  low  voltage,  it  was  thought  that  the  phenomena  of  the 
brush  discharge  and  electrostatic  convection  might  pertain  only 
to  the  extraordinary  voltages  of  the  so-called  electrostatic 
machines;  but  the  development  of  high  voltage  power  transmis- 
sion has  proved  that  those  phenomena  are  associated  with  vol- 
tages of  only  a few  tens  of  thousands  of  volts  and  that  they 
occur  in  striking  degree  on  aerial  power  lines  when  the  voltage 
exceeds  a hundred  thousand  or  more  volts.  Such  lines,  when 
the  voltage  exceeds  125,000  volts,  become  luminous  from  the 
discharge,  and  seem,  when  viewed  in  the  dark,  to  be  covered 
with  a brush  discharge  having  the  appearance  of  the  correspond- 
ing discharge  between  the  electrodes  of  an  electrostatic  machine. 

The  discharge  is  established  at  a voltage  which  is  inversely 
related  to  the  radius  of  curvature  of  the  conductors  on  which 
it  is  seated,  and,  in  the  case  of  aerial  lines  for  the  transmission 
of  power  by  alternating  currents,  it  results  in  a serious  loss  of 
power  when  the  voltage  much  exceeds  125,000  volts  and  the 
wires  are  of  the  usual  transmission  sizes  such  as  from  No.  6 to 
No.  0000  B.  and  S.  gauge.  The  loss  is  greater  when  the  baro- 
metric pressure  of  the  atmosphere  is  low,  and  it  is  therefore 
more  strikingly  observed  in  connection  with  high  voltage  lines 
in  mountainous  regions.  The  phenomena  thus  denoted  are 
called  Corona  effects,  on  account  of  the  luminous  corona  which 
may  be  observed  in  a dark  room  around  conductors  on  which 
the  phenomena  are  active.  The  phenomena  are  accompanied 


954 


ALTERNATING  CURRENTS 


by  the  crackling  or  hissing  sounds  and  the  production  of  nitric 
oxide  and  ozone  (which  may  be  observed  by  the  odor)  that  are 
recognized  accompaniments  of  brush  discharges  at  the  terminals 
of  an  electrostatic  machine. 

The  amount  of  power  lost  by  convection  between  power  trans- 
mission conductors  of  the  usual  diameters  increases  slightly 
with  the  voltage  up  to  a certain  critical  point,  at  which  the 
corona  seems  to  be  fuily  established,  and  the  ratio  of  power 
lost  to  line  voltage  is  much  higher  for  voltages  above  the  criti- 
cal point.  The  critical  voltage  for  ordinary  transmission  lines 
at  the  usual  central  altitudes  of  this  country  seems  to  be  about 
125,000  volts,  and  the  loss  would  be  prohibitive  on  long  lines 
at  higher  voltages  when  conductors  not  exceeding  one  half  inch 
in  effective  diameter  are  used.  The  loss  also  increases  with  the 
frequency  of  the  cycles  of  the  voltage. 

These  phenomena  have  been  given  much  study,  but  no  reli- 
able laws  have  yet  been  formulated.  The  consensus  of  knowl- 
edge may  be  obtained  fairly  well  from  various  papers  in  the 
Transactions  of  the  American  Institute  of  Electrical  Engineers 
for  the  years  1910  to  1912.* 

It  has  not  yet  been  conclusively  determined  whether  or  not 
the  effects  of  corona  may  occur  in  liquids  or  solids,  but  the 
best  opinion  leans  to  the  hypothesis  that  corona  is  a result  of 
ionization  in  a gas  and  requires  a gaseous  medium  for  its  devel- 
opment. There  are,  however,  certain  recognized  effects  of  ex- 
cessive electrostatic  stresses  in  solid  dielectrics  which  indicate 
that  some  phenomena  occur  in  solids  which  are  at  least  similar 
to  corona  effects. 

* Steinmetz,  Ryan,  Merslion,  Whitehead,  Peek,  and  others. 


INDEX 


A 

Active  current,  313. 

Active  voltage  locus  of  transformer, 
486. 

Adams,  operation  of  alternators  in 
parallel,  669. 

Admittance,  the  reciprocal  of  im- 
pedance, defined,  212  ; expressed  as 
a complex  quantity,  214. 

Admittances,  combination  of,  253-256. 

Ageing,  effect  of,  on  iron  in  transformers, 
509-510. 

Air  blast,  cooling  transformers  by,  543- 
544. 

Air-cooled  transformers,  539. 

Air  friction,  losses  in  alternators  due 
to,  597. 

All-day  efficiency  of  transformers,  516. 

.Alliance  dynamo,  90. 

Alloying  materials  used  in  transformers, 
515. 

Alternating  circuit,  current  in,  316- 
317  ; methods  for  measuring  power 
in,  344-358. 

Alternating  current,  controlled  by  same 
laws  as  direct  current,  1-2  ; relation 
between  alternating  voltage  and, 
2-3 ; period  and  frequency  of,  3 ; 
heating  effect  or  power  activity  of 
(Joule’s  law),  3 ; determination  of 
effective  value  of,  4-6 ; frequencies 
of,  20-22 ; instruments  for  measur- 
ing, 43-45,  65-75. 

Alternating-current  curve,  resolution 
of,  into  its  harmonic  components, 
35-42 ; determining  the  effective 
ordinate  of  an,  42-43. 

Alternating-current  generator,  19. 

Alternating-current  motor-generators, 
782-783. 

Alternating  - current  motors,  series, 
874-881  ; commutation  and  other 
characteristics  of  commutating,  887- 
891  ; vector  diagrams  of  series 
motors,  and  expressions  for  voltage, 
881-887. 

Alternating  flux,  caused  by  current  of 
predetermined  form,  438-440. 

Alternating  voltage,  2 ; instruments 
for  measuring,  43-45. 

Alternators,  19 ; form  of  voltage 
curve  of,  22-27 ; measurement  of 
effective  voltage  developed  by,  45 ; 
comparison  of  voltages  developed  by. 


and  by  direct-current  dynamo,  46 ; 
single-phase  and  polyphase,  55-60 ; 
armature  windings  for,  77-119  ; field 
excitation  of,  106-108 ; composite 
excitation  of,  108-114;  inductor, 
117-119;  calculation  of  effective 
value  of  voltage  of  armature,  218- 
220  ; losses  in,  597-600  ; relation  of 
speed  to  weight  in,  602-604 ; self- 
inductance of,  609-611;  character- 
istics of.  621  ; regulation  of,  for 
constant  voltage,  654-661 ; con- 
necting, for  combined  output,  664 ; 
in  series,  664-668 ; in  parallel,  668- 
669  ; synchronizing  current  of,  669- 
676 ; division  of  load  between 
parallel,  676-686  ; maximum  possible 
load  and  regulation  of  prime  movers 
in  parallel  operation,  686-688  ; effect 
of  form  of  voltage  curve  and  varia- 
tion of  angular  velocity  on  parallel 
operation  and  division  of  loads, 
689-690 ; methods  of  connecting, 
in  parallel  to  feeder  circuits,  699- 
702 ; as  synchronous  motors,  702- 
704 ; methods  of  testing,  729  ff. ; 
efficiency  by  rated  motor  and  stray 
power  measurements,  729-732  ; feed- 
ing-back method  for  measuring  effi- 
ciency, 732-733  ; efficiency  by  rated 
motor,  733-734 ; Mordey’s  method 
of  testing,  734-736 ; divided  field 
method,  736-737 ; motor-generator 
method,  737-740 ; heating  tests, 
740-741  ; insulation  tests,  741  ; regu- 
lation, 741-742 ; wave  form  of 
voltage  produced  by,  742. 

Amortisseurs  for  preventing  hunting, 
747. 

Amperemeter  (three)  method  of  measur- 
ing power,  356. 

Amperemeters,  alternating  current,  43- 
45,  65—75 ; arrangement  of,  for 

measuring  high  voltages  or  large 
currents,  76  ; alternating-current,  for 
testing  alternators,  730. 

Ampere  turns,  exciting,  446 ; of  ar- 
matures, 609. 

Amplitude  of  scalar  value  of  vectors,  11. 

Analogies  of  the  electric  circuit,  945- 
952. 

Analytical  method  for  solution  of 
problems,  261. 

Angle  of  lag,  132-133 ; method  of 
measuring,  343. 


956 


INDEX 


Apparent  power,  325. 

Argument  of  vector  quantity,  242. 

Armatures,  multipolar,  45  ; comparison 
of  voltages  developed  in  alternator 
and  direct-current,  46 ; effect  of 
arrangements  of  windings  on  voltages 
of,  47-54  ; classification  of,  77  ; in- 
sulating, 120-123 ; core  materials, 
123-124 ; ventilation  of,  124-126 ; 
measurement  of  self-inductance  of, 
145 ; curve  showing  relation  of 
current  to  excitation,  724-727  ; heat- 
ing of  conductors  in  rotary  converters, 
765-772;  reactions  of  converter, 
772-773  ; action  of  a short-circuited 
winding  within  rotating  field,  790- 
792 ; of  induction  motors,  791 ; 
coil-wound,  connected  to  external 
devices,  in  construction  of  rotating 
field  induction  motors,  825 ; re- 
sistance in  circuits  for  regulation  of 
speed  of  induction  motors,  835-837  ; 
commutated,  837-838. 

Armature  conductors,  determination 
of  number  of,  605-609. 

Armature  cores  and  conductors,  losses 
in  alternators  due  to  eddy  currents 
and  hysteresis  in,  597. 

Armature  reactions  in  alternators,  611- 
621. 

Armature  windings  for  alternators, 
77-119. 

Arnold,  E.,  80. 

Asynchronous  generator,  784 ; rotary 
field  induction  motor  as  an,  822-824. 

Asynchronous  motors,  784  ff. 

Attenuation  constant,  938. 

Automatic  devices  for  regulation  of 
alternators,  654-661,  699-702. 

Automatic  switches,  764-765. 

Automatic  synchronizers,  699. 

Autotransformers,  576-581 ; use  of, 
as  voltage  regulators,  663-664 ; for 
regulating  induction  motors,  833-835. 

Ayrton,  measurement  of  self-induct- 
ance, 145-146  ; measuring  power 
in  alternating-current  circuit,  356 ; 
testing  transformers,  650 ; testing 
alternators,  736. 

Ayrton  and  Sumpner,  three-voltmeter 
method  of  measuring  power  in  al- 
ternating-current circuit,  354 ; 
method  of  obtaining  transformer 
efficiency,  589. 

B 

Balanced  polyphase  system,  359 ; 
uniformity  of  power  in,  367-368 ; 
measurement  of  power  in,  389  ff. 

Ballistic  galvanometer,  effect  on  resist- 
ances of  shunting  a,  163—165 ; use 
of,  in  tracing  curves,  649. 


Barrel  winding,  86,  102. 

Bar  windings,  78. 

Bedell,  testing  alternators,  652-654. 

Bedell  and  Crehore,  Alternating  Cur- 
rents, cited,  301. 

Behrend,  relation  of  weight  to  speed  in 
alternators,  602. 

Blakesley,  T.  H.,  application  of  graphi- 
cal processes  by,  257  ; measurement 
of  power,  316 ; three-instrument 
method  of  measuring  suggested  by, 
356. 

Blondel,  measurement  of  power,  395; 
tracing  curves,  650. 

Blondel  and  Duddell,  oscillograph  of, 
644. 

Booster,  580. 

Brown,  C.  E.  L.,  parallel  operation  of 
alternators,  689. 

Brush  alternator,  91. 

Brushes  of  collectors,  103,  105. 

Byerly,  Fourier's  Series  and  Spherical 
Harmonics,  cited,  27. 

C 

Calculation  of  magnetic  leakage,  581- 
583. 

Capacity,  electrostatic,  168 ; unit  of, 
168 ; specific  inductive,  169 ; con- 
ditions of  establishment  and  termina- 
tion of  current  in  a circuit  containing, 
170-175 ; effect  of,  in  a circuit,  185- 
186 ; conditions  of  establishment 
and  termination  of  current  in  circuit 
containing  resistance,  inductance, 
and,  in  series,  187-199 ; vector 
relations  of  current  and  voltage  in 
circuit  containing  resistance,  self- 
inductance, and,  in  series,  209-211. 

Capacity  circuit,  definition,  279  n. 

Capacity  reactance,  212. 

Capacity  voltage,  177,  267. 

Characteristics,  alternator,  621. 

Charge  of  a condenser,  169. 

Choking  coils,  575. 

Chord-wound  alternators,  78. 

Chord-wound  armature,  methods  of 
applying  the  wires  to,  93-102. 

Circle,  power,  685. 

Circle  diagram,  for  non-reactive  sec- 
ondary circuit,  4S6 ; where  trans- 
former load  contains  constant  react- 
ance and  variable  resistance,  490- 
498  ; of  magnetic  fluxes  in  polyphase 
induction  motor,  816-822. 

Circuit,  effects  of  changing  resistance 
of  a,  166-167 ; effect  of  introducing 
resistance  in  a continuous  current, 
180-183 ; effects  of  self-inductance 
and  capacity  in,  183-186 ; effect 
on  transient  state  in  a,  of  self-induct- 


INDEX 


957 


ance  and  capacity  combined,  186- 
187 ; time  constant  of  a,  206-208. 
See  Self-inductive  circuit. 

Circuits,  solution  of,  by  graphical 
methods,  257  ff. ; classification  of, 
for  solution,  261-262  ; series,  262-277 ; 
parallel,  277-301  ; signification  of 
terms  inductive,  capacity,  reactive, 
and  non-reactive  circuits,  279  n. ; 
series  and  parallel  combined,  301- 
309 ; mutual  induction  of  parallel 
distributing,  911-916;  effects  of  self 
and  mutual  induction  in  polyphase, 
916-922  ; distortionless,  943  ; tuned, 
947. 

Circular  loci  for  constant  current 
transformer,  570. 

Classification  of  circuits,  for  solution  by 
graphical  methods,  261-262. 

Coefficient,  of  self-induction,  135 ; of 
mutual  induction,  450. 

Coils,  current,  345 ; voltage,  345 ; 
reactance  of,  causing  errors  in  watt- 
meter readings,  346-349 ; impedance, 
467,  575;  reactance,  467,  575; 

choking,  575 ; shading,  853 ; com- 
pensating, 876. 

Coil  windings,  78,  85. 

Coil-wound  armatures  connected  to 
external  devices,  in  construction  of 
rotating  field  induction  motors,  825. 

Coincidence  of  voltage  and  current 
phases,  325. 

Collector  rings,  19,  47,  102-106. 

Combination  of  admittances,  253-256. 

Commutated  armature,  837-838- 

Commutated  primary  windings,  838. 

Commutation  of  commutating  alternat- 
ing-current motors,  887. 

Commutator,  rectifying,  114-117. 

Compensated  alternator,  113. 

Compensating  coils,  876. 

Compensators,  576-581  ; variable,  for 
regulating  induction  motors,  833- 
835. 

Complementary  vectors,  246. 

Complex  expressions,  addition  and 
subtraction  of,  244 ; multiplication 
and  division  of,  244-246. 

Complex  quantities,  17 ; involution 
and  evolution  of,  249-251  ; differen- 
tiation and  integration  of,  251-252  ; 
logarithms  of,  252-253. 

Complex  quantity,  vector  analysis  and 
the,  241-244. 

Composite  excitation  of  alternators, 
107-114. 

Compound-wound  machines,  108,  110. 

Concatenation  control,  induction  mo- 
tors, 838-840. 

Condenser,  the  term,  169 ; dielectric 
of  a,  169 ; the  charge  of  a,  169 ; I 


curves  of  charge  and  discharge  of  a, 
171-173;  energy  of  a charged,  175- 
176 ; synchronous,  748-754. 

Condenser  circuit,  definition,  279  n. 

Condensers,  total  impressed  voltage  of 
circuit  when  connected  in  series,  276 ; 
current  when  connected  in  parallel, 
297-298. 

Condenser  voltage,  177,  267. 

Conductance,  of  a circuit,  214 ; of  an 
admittance,  253. 

Conductively  compensated  series  motor, 
877. 

Conductor  losses  in  transformers,  64. 

Conductors,  density  of  current  in,  of 
alternators,  602-604 ; electrostatic 
capacity  of  parallel,  922-931. 

Connecting  constant  voltage  trans- 
formers, 546-561. 

Connecting  up  armature  windings, 
58-60. 

Constant  current  transformers,  468,  567. 

Constant  voltage  transformers,  468 ; 
constructive  features  of,  531-546. 

Constants,  determination  of  value  of, 
in  Fourier’s  Series,  33-35;  hys- 
teresis, 408-409. 

Construction  of  rotating  field  induction 
motors,  824-832. 

Contact  makers,  593 ; methods  of 
using,  in  determining  form  of  voltage 
or  current  curves,  648-652. 

Converters,  rotary,  83,  754-758 ; ratio 
of  transformation  in,  758-763 ; fre- 
quency and  voltage  limitations  in, 
763-765  ; heating  of  armature  con- 
ductors in  rotary,  765-772 ; arma- 
ture reactions  of,  772-773 ; voltage 
control  of,  773 ; split-pole,  773 ; 
synchronous  regulators,  774  ; features 
of  operation,  775-781  ; inverted, 
781-782  ; mercury  vapor,  895-896. 

Cooling,  of  transformers,  538-544 ; 
of  alternator  armatures,  600. 

Copper-band  contactors,  105. 

Copper  conductors,  self-inductances  of 
solid  cylindrical,  901. 

Copper  losses  in  transformers,  519-520. 

Core  magnetization,  483-486. 

Core  materials  in  armatures,  123-124. 

Core  type  transformers,  532-534. 

Corona  effects,  953-954. 

Counter  voltage,  61  ; induced  in  pri- 
mary circuit  of  induction  motor,  792- 
795. 

Current,  conditions  of  establishment 
and  termination  of,  in  circuit  con- 
taining resistance  and  self-inductance 
in  series,  146-149 ; effect  on  rise 
and  fall  of,  of  eddy  currents  and  of 
hysteresis,  165-166 ; establishment 
and  termination  of,  in  circuit  con- 


958 


INDEX 


taming  capacity  and  resistance  in 
series,  170-175;  establishment  and 
termination  of,  in  a circuit  containing 
resistance,  inductance,  and  capacity 
in  series,  187-199 ; measure  of  the 
growth  of,  by  the  time  constant,  206- 
207 ; 'general  equation  for,  in  a cir- 
cuit, 222-224  ; in  a circuit  when  any 
periodic  voltage  is  impressed,  224- 
226 ; irregular,  and  voltage  waves 
expressed  as  complex  quantities,  226- 
229 ; envelope  of  vector  when  condi- 
tions in  circuit  vary,  237-240 ; in 
series  circuits  under  varying  condi- 
tions, 276 ; in  parallel  circuits,  297- 
298;  active  or  energy,  313;  wattless 
or  quadrature,  314 ; magnetizing, 
476  ; curves  of  voltage  and,  634  ff. ; 
distribution  of,  in  a wire,  907-911. 

Current  coil,  345. 

Current  curve,  methods  for  deter- 
mining form  of,  644-652. 

Current  rushes,  583-584. 

Current  surges,  583-584. 

Current  transformers,  571-575. 

Currents,  relations  between  voltages 
and,  in  polyphase  systems,  370  ff. ; in 
rotary  field  induction  motor,  800-803. 

Curve,  alternating  voltage,  2 ; of 
squared  instantaneous  ordinates,  4 ; 
harmonic,  produced  by  rotating 
vector,  9-10  ; form  of  voltage,  of  an 
alternator,  22-27  ; method  of  finding 
the  harmonics  of  a periodic,  by 
Fourier’s  Series,  35-38  ; of  voltage  in 
separate  slots  of  progressive  distrib- 
uted windings,  48,  50,  51  ; satura- 
tion, or  magnetization,  621-624 ; 
external  characteristic,  621,  624-632 ; 
of  synchronous  impedance,  621, 
632-634  ; magnetic  distribution,  621, 
634  if. ; voltage,  621,  634  ff. ; short- 
circuit  current,  632 ; showing  rela- 
tion of  armature  current  to  excita- 
tion, 724-727. 

Curves,  equations  of,  by  Fourier’s 
Series,  27-33  ; table  of  characteristic 
features  of  different  forms  of  alternat- 
ing-current and  voltage,  43  ; of  rise  of 
charge  and  discharge  in  a condenser, 
171-173  ; of  hysteresis,  and  of  current 
and  voltage  distorted  by  hysteresis, 
404-407 ; of  hysteresis  and  eddy 
current  losses  in  transformer  irons 
and  steels,  509-515 ; areas  of  succes- 
sive, of  alternating  currents  and  vol- 
tages, 652-654. 

Curves  of  torque  of  induction  motor, 
857,  858. 

Curves  of  voltage,  effect  of  form  of,  on 
operation  of  induction  motors,  842- 
843. 


Cycles  of  frequencies  of  alternating 
currents,  20-21. 

Cyclic  curves,  of  eddy  currents,  425 ; 
of  iron  loss,  476-477. 

Cyclic  hysteresis  curves,  404. 

Cylindrical  conductors,  eddy  current 
loss  in,  418-421. 

D 

Damping  grids,  747. 

De  la  Tour,  Moteurs  Asynchronous 
Polyphases,  cited,  52. 

Delta  connection,  368-370,  552-554. 

Delta  winding,  57,  364. 

De  Meritens  armature,  92. 

Deptford  Station  transmission  plant, 
91,  105. 

Dielectric  constant,  the,  169. 

Dielectric  hysteresis,  407-408. 

Dielectric  of  a condenser,  169. 

Dielectric  strength,  of  insulating  ma- 
terials, 121-123 ; of  transformer 
insulation,  592-593. 

Dielectric  strength  tests  of  alternators, 
741. 

Differentiation  and  integration  of  com- 
plex quantities,  251. 

Dimmer,  580. 

Direct-current  dynamo,  conversion  into 
a double-current  machine,  82-83. 

Direction  coefficient  of  an  expression, 
242. 

Disk  armatures,  90-92. 

Distortionless  circuit,  943. 

Distributed  coils,  27 ; effect  of,  on 
armature  voltage,  47-54. 

Distributed  resistance,  932. 

Distributed  windings,  54,  78,  85-90. 

Distribution  of  current  in  a wire,  907- 
911. 

Divided  circuits,  transient  transfer  of 
electricity  in,  162-165. 

Divided  field  method  of  measuring  al- 
ternator losses  or  efficiency,  736- 
737. 

Dobrowolsky,  784. 

Double-current  generators,  782. 
i Double-current  machine,  47. 

I Double  delta  connection,  563. 

Drehfelde,  790. 

Drehstrom,  790. 

Drum  winding,  78. 

Dry-core  transformers,  539. 

Du  Bois,  H.,  electro-magnet  designed 
by,  145. 

Duncan,  measurements  by,  145 ; testing 
transformers,  594 ; tracing  curves, 
649. 

Dynamo,  voltage  developed  by  a direct- 
current,  compared  with  that  de- 
veloped by  an  alternator,  46  ; convert- 


INDEX 


959 


ing  direct-current  into  double-current 
machine,  82-83. 

Dynamometer,  measuring  power  by  a 
split,  357-358. 

E 

Eddy  currents,  effect  of,  on  rise  and 
fall  of  current  in  self-inductive 
circuit,  165-166 ; effect  of,  in  watt- 
meter frame,  350-351  ; cyclic  curves 
of,  425-427  ; magnetic  screening  due 
to,  427-433 ; effect  of,  on  form  of 
primary  current  wave,  480-481. 

Eddy  current  losses,  in  core  of  a trans- 
former, 64  ; from  magnetic  hysteresis, 
400-404  ; measurement  of,  410-417  ; 
computations  of,  417-424;  in  cylin- 
drical conductors,  418-421  ; in  sheets 
of  rectangular  cross  section,  421-424  ; 
curves  of,  in  transformer  irons,  512- 
515;  in  transformers,  518-519;  in 
alternators,  597. 

Effective  value,  of  alternating  current 
and  voltage,  4-6 ; of  an  irregular 
curve  of  voltage  or  current,  42-43. 

Effective  volts  and  amperes,  measure- 
ment of,  43-45. 

Efficiencies  of  transformers,  515-523. 

Efficiency,  of  alternators,  729  ff. ; of 
converters,  780-781. 

Electrical  degrees,  21. 

Electrical  resonance,  946. 

Electric  circuit,  analogies  of  the,  945- 
952. 

Electricity,  transference  of,  149-152 ; 
transient  transfer  of,  in  divided  cir- 
cuits, 162-165. 

Electro-dynamometers,  65-71,  344-346, 
357. 

Electrolytic  rectifiers,  896. 

Electromagnetic  instruments  for  meas- 
uring alternating  voltages  and  cur- 
rents, 43-44. 

Electromagnetic  repulsion,  863-874. 

Electrometer,  quadrant,  73-74. 

Electrometer  method  for  measuring 
power  in  alternating  circuit,  351-353. 

Electro-motive  force  of  self-induction, 
128. 

Electrostatic  capacity,  of  parallel  con- 
ductors, 922-931  ; relations  of  resist- 
ance, self-inductance,  and,  932  ff. 

Electrostatic  instruments  for  measur- 
ing alternating  voltages  and  currents, 
44-45,  73-74. 

Electrostatic  wattmeter,  for  measuring 
power  in  alternating  circuit,  353-354. 

Emery,  alternating-current  curves,  638. 

Energy,  stored  in  a magnetic  field 
associated  with  an  electric  circuit, 
153-162;  of  a charged  condenser, 


175-176;  of  mutual  induction,  453- 
456. 

Energy  current,  313. 

Energy  losses  caused  by  magnetic 
hysteresis,  400-404. 

Equalizer  connections  between  sections 
of  armature  windings,  779. 

Equations  of  curves  by  Fourier’s  Series, 
27-33. 

Equivalent  impedances,  use  of,  in 
solving  transformer  problems,  501- 
502. 

Equivalent  resistance  and  reactance, 

221. 

Equivalent  resistance  of  a circuit,  441- 
442. 

Equivalent  sinusoids,  319. 

Ewing,  .1.  A.,  experiments  by,  403,  404, 
407  ; hysteresis  tester,  414;  table  of 
ratio  of  magnetic  force  in  a plate  to 
magnetic  force  at  surface,  430-431; 
experiments  by,  to  determine  iron 
losses  in  transformers,  595-596. 

Ewing  and  Klaasen,  cited,  596. 

Ewing’s  apparatus,  596. 

Excitation,  of  alternators,  106-114; 
relation  of  armature  current  to,  724- 
727. 

Excitation  angle,  481. 

Exciters,  20,  106. 

Exciting  ampere  turns,  446. 

Exciting  current,  form  of,  required  to 
set  up  a given  core  magnetization, 
433-437  ; of  transformer,  475-477  ; 
for  an  induction  motor,  796-797. 

Exponential  terms,  200-204. 

External  characteristic,  alternator,  621, 
624-632. 

F 

Farad,  the  unit  of  capacity,  168. 

Feeder  circuits,  connecting  alternators 
in  parallel  to,  699-702. 

Feeder  regulators,  661-664. 

Feeding-back  method,  of  testing  trans- 
formers, 589  ; for  measuring  efficiency 
of  alternators,  732-733. 

Ferranti  alternator,  90,  91. 

Ferraris,  784. 

Field  current,  wavy  alternator,  116,  117. 

Field  excitation,  relation  of  armature 
current  to,  724-727. 

Field  frequency  of  induction  motor,  797. 

Field  magnet,  defined,  791. 

Field  windings  for  armatures,  77  ff. 

Fleming,  Alternate  Current  Transformer 
in  Theory  and  Practice,  cited,  33 ; 
transformer  tests,  583  ; determining 
iron  losses,  594-595 ; Propagation  of 
Electric  Currents  in  Telephone  and 
Telegraph  Conductors  by,  945  n. 


960 


INDEX 


Form  factor,  of  current  curve,  7 ; de- 
pendence of  iron  losses  of  transformer 
on,  506. 

Foucault  currents.  See  Eddy  currents. 

Fourier’s  Series,  equations  of  curves  by, 
27-33 ; determination  of  the  value 
of  the  constants  in,  33-35 ; finding 
the  harmonics  of  a periodic  curve  by, 
35-38. 

Frequencies,  commercial,  20-22. 

Frequency,  of  alternating  current,  3 ; 
effects  of  changes  of,  on  transformers, 
507-509 ; in  converters,  763-765 ; 
effect  of,  on  induction  motors,  859- 
860. 

Frequency  changer,  782,  891. 

Frequency  converter,  892-893. 

Frequency  indicator,  782. 

Friction,  similarity  between  magnetic 
and  dielectric  hysteresis  and,  407-408  ; 
losses  in  alternators  due  to,  597. 

Functions,  alternating  and  pulsating, 
319-320. 

G 

Galvanometer,  effect  of  shunting  a 
ballistic,  163-165. 

Ganz  and  Co.,  alternator,  115-116. 

Generators,  synchronous,  19 ; multi- 
polar, 20-22  ; double-current,  782  ; 
asynchronous,  784. 

Gerard,  cited,  354 ; device  for  tracing 
voltage  curves  of  an  alternator,  647. 

Gilbert,  defined,  400. 

Graphical  indication  of  effect  of  induc- 
tive load,  626-628. 

Graphical  solutions  of  problems,  257  ff . ; 
in  series  circuits,  262-275;  in  series 
circuits,  conclusions,  277  ; in  parallel 
circuits,  277  If. ; in  parallel  circuits, 
conclusions,  297-299 ; in  series  and 
parallel  circuits  combined,  302-309. 

Gray,  distribution  of  current  in  a wire, 
910. 

H 

Harmonics,  spherical,  28-33 ; method 
of  finding,  of  periodic  curve,  35-38 ; 
variation  in  impedance  offered  to 
current  and  voltage,  of  different 
frequencies,  229-236 ; effect  of,  in 
waves  of  voltage  and  current  upon 
operation  of  a transformer,  505-507. 

Harmonic  voltages  and  currents,  11. 

Haselwander,  784. 

Hay,  investigation  of  current  rushes 
by,  583. 

Hayward,  Vector  Algebra  and  Trigo- 
nometry, cited,  243. 

Heating,  testing  transformers  as  to, 
591-592 ; tests  of  alternators  by. 


740-741  ; of  armature  conductors 
in  rotary  converters,  765-772. 

Heating  effect  of  alternating  current,  3. 

Heaviside,  Electrical  Papers,  cited,  931. 

Hedgehog  transformer,  517. 

Henry,  defined,  136,  452. 

Hobson,  Plane  Geometry,  cited,  49. 

Hoffman,  tracing  curves,  649. 

Holden,  hysteresis  tester,  415—416. 

Hopkinson,  John,  594 ; operation  of 
alternators  in  parallel,  669. 

Hospitalier,  cited,  354. 

Hot  wire  instruments  for  measuring 
alternating  voltages  and  currents, 
44,  71-73. 

Houston  and  Kennedy,  article  by,  33  n. 

Hunting  of  synchronous  machines, 
743-748. 

Hutchinson,  tracing  curves,  649. 

Hysteresis,  effect  of,  on  rise  and  fall  of 
current  in  a self-inductive  circuit, 
165-166;  energy  losses  caused  by 
magnetic,  400-404  ; curves  of,  404- 
407 ; similarity  between  magnetic 
and  dielectric,  and  friction,  407-408 ; 
constants  of,  408-409  ; measurement 
of  energy  absorbed  by,  410—417 ; 
form  of  exciting  current  in  circuit 
with  and  without,  433-437 ; effect 
of,  on  form  of  primary  current  wave, 
479-480. 

Hysteresis  loop,  403. 

Hysteresis  losses,  in  core  of  a trans- 
former, 64 ; measurement  of,  410- 
417  ; curves  of,  in  transformer  irons, 
510-512;  in  transformers,  518-519; 
in  alternators,  in  armature  cores,  597. 

Hysteresis  testers,  414-416. 

Hysteretic  angle  of  advance,  435,  442, 
481. 

I 

Impedance,  of  alternating-current  cir- 
cuit, 149,  211-214;  in  circuit  con- 
taining capacity  and  resistance  in 
series,  175 ; in  circuit  containing 
resistance,  inductance,  and  capacity 
in  series,  197;  polygons  of,  212; 
expressed  as  a complex  quantity, 
213-214 ; variation  in,  offered  to 
current  and  voltage  harmonics  of 
different  frequencies,  229-236 ; syn- 
chronous, 633,  706-715;  in  rotary 
field  induction  motor,  800-801 ; sub- 
stituted, for  rotary  field  induction 
motor,  807-815;  locus  diagram  of 
single-phase  motor  and  substituted, 
850-852. 

Impedance  coils,  467,  575. 

Impedance  methods,  solution  of  parallel 
circuits  by,  299—301 : solution  of 

series-parallel  problems  by,  309-310. 


INDEX 


961 


Impressed  voltage,  129 ; in  self-in- 
ductive and  capacity  circuits,  179- 
180 ; solution  of  series-parallel  prob- 
lems by  method  of,  309-310. 

Indicator,  power-factor,  743;  frequency, 
782. 

Inductance,  conditions  of  establish- 
ment and  termination  of  current  in 
circuit  containing  resistance,  capacity, 
and,  in  series,  187-199  ; mutual,  449- 
453  ; leakage,  468. 

Induction,  mutual,  of  parallel  dis- 
tributing circuits,  911-916;  effects 
of  self  and  mutual,  in  polyphase  cir- 
cuits, 916-922. 

Induction  coils,  61-62 ; primary  and 
secondary,  62. 

Induction  factor,  343. 

Induction  instruments  for  measuring 
alternating-current  circuits,  75. 

Induction  motor,  65,  784,  791  ; rotary 
field,  784-785  ; counter  voltage  in- 
duced in  primary  circuit  of,  792- 
795  ; exciting  current  for,  796-797  ; 
speeds,  797-798  ; slip,  798  ; secondary 
induced  voltage,  798-800 ; currents, 
torque,  impedance,  and  magnetic 
leakage  in  a rotary  field,  800-803 ; 
vector  relations  in  the  rotating  field, 
803-807  ; substituted  impedance  for 
the  rotary  field,  807-815;  formulas 
for  torque  and  slip  of  a polyphase, 
815  ff. ; circle  diagram  of  magnetic 
fluxes,  816-822  ; the  rotary  field,  as 
an  asynchronous  generator,  822-824; 
squirrel-cage  form  of  windings,  824 ; 
independent  short-circuited  coils,  824; 
features  of  construction  of  rotating 
field,  824  ff. ; starting  and  regulating 
devices,  832  ff. ; reversing  polyphase, 
840-842 ; effect  of  form  of  curves 
of  voltage  on  operation  of,  842-843  ; 
single-phase,  843-850  ; locus  diagram 
of  single-phase,  and  substituted  im- 
pedance, 850-852 ; starting  single- 
phase, 852-853 ; efficiency  of,  and 
methods  of  making  tests,  853-859 ; 
effect  of  frequency,  859-860 ; poly- 
phase, with  exciting  current  supplied 
to  the  armature,  860-863. 

Inductive  circuit,  definition,  279  n. 

Inductive  circuits  of  equal  time  con- 
stants, connected  in  series,  276 ; 
connected  in  parallel,  297. 

Inductively  compensated  series  motor, 
877. 

Inductive  reactance,  212 ; effect  of, 
in  secondary  external  circuit,  490- 
492. 

Inductor  alternators,  77,  117-119. 

Instruments  for  measuring  alternating 
voltages  and  currents,  43-45. 

3 Q 


Insulation,  of  armatures,  120-123 ; 
dielectric  strength  of  insulating 
materials,  121-123 ; in  transformers, 
544-  545  ; dielectric  strength  of  trans- 
former insulation,  592-593. 

Insulation  Tests  of  alternators,  741. 

Inverted  converters,  755,  781-782. 

Involution  and  evolution  of  complex 
quantities,  249-251. 

Iron,  constants  of  magnetic  hysteresis 
related  to,  409  ; quality  of,  for  trans- 
formers, 509-515. 

Iron  losses,  221,  412  ; due  to  hysteresis, 
402-404;  measurement  of,  410-414; 
cyclic  curve  of,  425-427 ; effect  of, 
on  apparent  resistance  and  reactance 
of  a circuit,  440-442 ; cyclic  curve  of, 
476-477 ; dependence  of,  upon  form 
factor,  506 ; in  transformers,  519- 
520 ; experiments  to  determine,  in 
transformers,  594-596 ; in  alter- 
nators, 599-600. 

Irregular  current  and  voltage  waves 
expressed  as  complex  quantities, 
226-229. 

Irregular  rotary  field,  787-788. 

Irregular  waves,  examples  of,  28-32. 

J 

Jackson,  Electromagnetism  and  Con- 
struction of  Dynamos , cited,  23,  33, 
47,  107,  120,  128,  135,  604,  732; 
Elementary  Electricity  and  Magnetism, 
cited,  43,  65,  68;  paper  on  “Three- 
phase  Rotary  Field,”  cited,  794. 

Joubert,  tracing  curves,  91  ; use  of 
contact  maker  by,  to  determine  form 
of  voltage  and  current  curves,  649. 

Joule’s  law,  3. 

K 

Kapp,  Dynamos,  Alternators,  and  Trans- 
formers, cited,  48,  123. 

Kapp  alternators,  armature  cores  in, 
123. 

Kelvin,  alternating  kilowatt  balance, 
71  ; electrostatic  voltmeter,  73-74. 

Kennelly  and  McMahon,  935. 

Kilo-volt-amperes,  325. 

Kirchoff’s  law  of  current  flow,  366. 

Kirchoff’s  Laws,  262. 

L 

Lag  angle,  132-133  ; measuring,  343  ; 
relation  of  wattmeter  readings  to,  in 
balance  three-phase  circuit,  397-399. 

Laminated  core  of  transformer,  64. 

Lamp  synchronizers,  691-694. 

Lap  windings,  80,  102. 


902 


INDEX 


Lead  and  lag,  11,  129. 

Leakage,  magnetic,  446,  451  ; magnetic, 
in  transformers,  465-468  ; calculation 
of  magnetic,  581-583  ; in  rotary  field 
induction  motor,  800. 

Leakage  conductance,  932. 

Leakage  flux,  451. 

Leakage  inductance,  468. 

Leakage  reactance,  468. 

Lenz's  Law,  128. 

Liquid  rheostats,  833. 

Load,  maximum  possible,  in  parallel 
operation  of  alternators,  686-687. 

Load  reactances  of  transformers,  498- 
501. 

Locus  diagrams,  of  currents,  voltages, 
and  loads  of  the  synchronous  motor, 
715-724  ; of  single-phase  motor  and 
substituted  impedance,  850-852 ; 
testing  induction  motors,  854. 

Locus  of  current  in  a transformer  when 
load  reactance  is  varied  and  resist- 
ance is  kept  constant,  498-501. 

Logarithms  of  complex  quantities, 
252-253. 

Loppe  et  Bouquet,  cited,  301,  423. 

M 

McAllister,  Alternating  Current  Motors, 
cited,  727,  889 ; vector  relations  in 
rotary  field  induction  motor,  805. 

McMahon,  Hyperbolic  Functions,  cited, 
929,  935,  936,  938. 

Magnetic  distribution  curve,  621,  634  ff. 

Magnetic  field,  energy  of  self-induced, 
153-162;  rotating,  619,  785-790. 

Magnetic  flux  distribution  curve  of  an 
alternator,  634  ff. 

Magnetic  fluxes,  circle  diagram  of,  in 
polyphase  induction  motor,  816-822. 

Magnetic  hysteresis,  energy  losses 
caused  by,  400-404. 

Magnetic  leakage,  446,  451 ; in  trans- 
formers, 465-468  ; diagram  of  trans- 
former with,  468-475;  calculation 
of,  581-583  ; in  rotary  field  induction 
motor,  800. 

Magnetic  linkages,  152 ; energy  stored 
in,  158-160. 

Magnetic  screening,  due  to  eddy  cur- 
rents, 427-133. 

Magnetic  vane  instruments,  74-75. 

Magnetism  wave,  relation  between 
form  of,  and  form  of  induced  voltage, 
437—138. 

Magnetization,  core,  483-486 ; curves 
of,  514,  621-624. 

Magnetizing  current,  476. 

Magnetomotive  force,  427. 

Mathematics,  employment  of,  by 
engineers,  241-242. 


Maxwell,  Electricity  and  Magnetism  by, 
cited,  902,  906,  909. 

Measurement,  of  alternating  voltages 
and  currents,  43-45 ; of  self-induct- 
ance, 136-141  ; of  power  factor, 
343  ; of  angle  of  lag,  343  ; of  power 
in  alternating  circuit,  344-358 ; of 
power  in  polyphase  systems,  389- 
399  ; of  energy  absorbed  by  hystere- 
sis, 410-417. 

Mechanical  analogue  of  an  electric 
circuit,  947-948. 

Mellor,  J.  W.,  Higher  Mathematics, 
cited,  33. 

Mercury  vapor  rectifier,  895-896. 

Merriman  and  Woodward,  Higher 
Mathematics,  cited,  33. 

Merritt,  testing  transformers,  594. 

Mershon,  testing  transformers,  650. 

Mesh  winding,  57,  364. 

Mhos,  unit  for  measuring  admittance, 

212. 

Microfarad,  the,  168. 

Mordey,  testing  alternators,  594,  734- 
736,  737. 

Mordey  alternator,  90,  91. 

Motor-converter,  893-895. 

Motor-generator,  782-783,  891-892. 

Motor-generator  method  for  measuring 
efficiency,  losses,  and  heating  of  an 
alternator,  737. 

Motors,  alternators,  as  synchronous, 
702-704 ; relation  of  voltages,  cur- 
rents, and  power  in  synchronous, 
704-715;  asynchronous,  784;  series, 
784  ; induction,  784,  791  ; repulsion, 
784,  863-874  ; rotary  field  induction, 
784-785;  series  alternating  current. 
874-881 ; inductively  and  conduc- 
tively  compensated  series,  877  ; com- 
mutation of  commutating  alter- 
nating-current, 887. 

Multiphase  machines,  55-58. 

Multipolar  armature,  45 ; with  wind- 
ing distributed  in  four  slots,  48,  49, 
51. 

Multipolar  generators,  20-22. 

Murray,  Differential  Equations,  cited, 
148, "l74,  181,  188,  190.  191,  196,  222, 
460. 

Mutual  inductance,  449-453. 

Mutual  induction,  62,  443-449  ; coeffi- 
cient of,  450  ; energy  of,  453—156  ; 
transfer  of  electricity  by  the  effect  of, 
456-463 ; of  parallel  distributing 
circuits,  911-916. 

N 

National  alternator,  93. 

Neutral  point,  56;  of  wye  connection, 
365. 


INDEX 


963 


Non-inductive  circuits,  connected  in 
series,  276  ; power  loops  in,  320. 

Non-reactive  circuits,  279  ; current  in, 
when  connected  in  parallel,  297 ; 
expenditure  of  power  in,  312. 

O 

Oersted,  discovery  by,  1. 

Oil,  transformer,  539. 

Oil-cooled  transformers,  539. 

One-circuit  windings,  82. 

Open-circuit  current,  488. 

Open  delta  connection,  552-554. 

Opposition  method  of  testing  trans- 
formers, 589. 

Oscillograph,  593-594,  644-648. 

Over-compensated  series  motor,  878. 

Over-compounded  machines,  108,  110. 

P 

Parallel,  mutual  inductance  in  coils  in, 
459-463  ; alternators  in,  668-669  ; 
division  of  load  between  alternators 
in,  676-686 ; maximum  possible 
load  and  regulation  of  prime  movers 
in  operation  of  alternators  in,  686- 
688. 

Parallel  circuits,  261 ; solutions  of 
problems  in,  277-299 ; solution  of, 
by  the  impedance  methods,  299-301 ; 
solution  of  combined  series  and,  302- 
309. 

Parallel  conductors,  electrostatic  capac- 
ity of,  922-931. 

Parallel  distributing  circuits,  mutual 
induction  of,  911-916. 

Parallel  operation,  of  alternators,  668- 
669  ; effect  of  form  of  voltage  curve 
and  variation  of  angular  velocity  on, 
689. 

Parallel  wires,  self-inductance  of,  897- 
907. 

Parshall  and  Hobart,  diagrams  of  al- 
ternator windings,  80  n. 

Pender,  computing  formulas,  935  n. 

Pender  and  Osborne,  electrostatic  capac- 
ity of  parallel  conductors,  929-930. 

Period  of  alternating  current,  3. 

Permeability,  effect  of,  on  self-induct- 
ance, 137-138 ; curve  of,  of  trans- 
former steels,  514-515. 

Perry,  John,  article  by,  33  n. 

Phase,  8. 

Phase  diagram,  12,  260,  443-449. 

Phase  difference,  9. 

Phase  indicators,  696-699. 

Phase  splitter,  852. 

Phase  transformation,  561-566. 

Picou,  quoted,  116. 

Plant  efficiency,  339. 


Polar  coordinates,  13-15. 

Polar  curve,  method  of  obtaining  effec- 
tive voltages  from  the,  13-15. 

Polarity  of  windings,  determination  of, 
547-548. 

Polar  surfaces,  46. 

Pole  armature,  92-93. 

Pole  face,  width  of,  54-55. 

Polygons,  of  voltages,  131,  149,  179, 
209-210 ; of  impedance,  212 ; of 
vectors,  260. 

Polyphase  alternators,  55-60  ; method 
of  connecting  up,  58-60 ; method  of 
compounding,  113. 

Polyphase  circuits,  effects  of  self  and 
mutual  induction  in,  916-922. 

Polyphase  induction  motor  with  ex- 
citing current  supplied  to  the  arma- 
ture, 860-863. 

Polyphase  machines,  58. 

Polyphase  systems,  defined,  359  ; bal- 
anced, 359  ; uniform  power  in,  367- 
368 ; relations  between  currents 
and  voltages,  370  ff. ; measurement  of 
power  in,  389-399. 

Polyphase  transformers,  524-531. 

Potentiometer  arrangement  for  contact 
maker,  651. 

Power,  expended  in  a circuit  on  which 
a sinusoidal  voltage  is  impressed, 
312-315;  in  alternating  circuit,  316; 
expended  in  circuit  when  voltage 
and  current  are  single-valued  periodic 
functions,  317-319;  apparent,  325; 
true,  325  ; vector,  333  ; quadrature, 
334 ; methods  for  measuring  in  al- 
ternating circuit,  344-358  ; measure- 
ment of,  in  polyphase  systems,  389- 
399. 

Power  circle,  685. 

Power  curves,  produced  by  sinusoidal 
voltages  and  currents  are  double 
frequency  sinusoids  wdth  axes  dis- 
placed, 323-324. 

Power  factor,  325 ; method  of  measur- 
ing, 343 ; relation  of  wattmeter 
readings  to,  in  balanced  three-phase 
circuit,  398-399 ; relation  of,  to 
capacity  in  transformers,  498-501  ; 
relation  of,  to  capacity,  500-501  ; of 
induction  motors,  856. 

Power-factor  indicator,  743. 

Power  factors,  table  of  reactive  factors 
and,  342. 

Power  loops,  320-323 ; quadrature 
components  of,  340-341. 

Power  relations,  expression  of,  by  means 
of  vectors,  333-339. 

Pressure  wires,  657. 

Primary  current  wave,  effects  of  vari- 
able reluctance,  of  hysteresis,  of 
eddy  currents,  and  of  current  in  the 


9G4 


INDEX 


secondary  winding  upon  form  of, 
477-483. 

Primary  windings,  791  ; commutated, 
for  regulating  induction  motors, 
838. 

Prime  movers,  regulation  of,  in  parallel 
operation  of  alternators,  686-688. 

Progressive  windings,  81-82,  102. 

Propagation  constant,  939. 

Pupin,  testing  transformers,  650. 

Q 

Quadrant  electrometer,  73,  351. 

Quadrature  components  of  power  loops, 
340-341. 

Quadrature  current,  314. 

Quadrature  power,  334. 

Quarter-phase  currents,  359-361. 

Quarter-phase  transformers,  524. 

R 

Rated  motor,  testing  alternators,  729- 
734. 

Ratio  of  transformation  in  a trans- 
former, 463-465. 

Rayleigh,  article  by,  cited,  909. 

Reactance,  of  alternating-current  cir- 
cuit, 149,  197,  212 ; of  circuit  con- 
taining capacity  and  resistance  in 
series,  175;  capacity,  212  ; inductive, 
212  ; expressed  as  a complex  quan- 
tity, 213-214  ; equivalent  or  working, 
221  ; of  coils,  errors  in  wattmeter 
readings  due  to,  346-349  ; of  a cir- 
cuit, effect  of  iron  losses  on  the  ap- 
parent, 440-442  ; leakage,  468. 

Reactance  coils,  467,  575. 

Reactions,  alternator  armature,  611- 
621  ; armature,  of  a converter,  772- 
773. 

Reactive  circuit,  definition,  279  n. ; 
expenditure  of  power  in,  312-313 ; 
power  loops  in,  320-321. 

Reactive  factor  of  a circuit,  342-343. 

Reactive  factors,  table  of  power  factors 
and,  342. 

Reactive  volt-amperes,  334. 

Rechniewski,  experiments  of,  600. 

Reciprocal  of  a vector  expression,  248. 

Rectifier,  mercury  vapor,  895-896  ; 
electrolytic,  896. 

Rectifying  commutator,  114-117. 

Regulation,  of  transformers,  522-523 ; 
of  alternators,  654-661,  741-742; 

of  induction  motors,  832  ff.,  856. 

Regulation  tests  for  transformers,  590- 
591. 

Regulators,  voltage,  575-581 ; feeder, 
661-664  ; synchronous,  774. 

Relays,  table  of  polarized,  143. 


Reluctance,  effect  of  variable,  on  form 
of  primary  current  wave,  477—479. 

Repulsion,  electromagnetic,  863-874. 

Repulsion  motors,  784 ; series,  878- 
879. 

Residual  magnetism,  effect  of,  on  rate 
of  change  of  magnetic  field  in  self- 
inductive  circuit,  166. 

Resistance,  effect  of  introducing,  in  a 
continuous  current  circuit,  180-183  ; 
conditions  of  establishment  and 
termination  of  current  in  circuit 
containing  inductance,  capacity,  and, 
in  series,  187-199 ; vector  relations 
of  current  and  voltage  in  circuit 
containing  self-inductance,  capacity, 
and,  in  series,  209-211  ; equivalent 
or  working,  221  ; effect  of  iron  losses 
on  apparent,  of  a circuit,  440 ; in 
field  circuit  of  induction  motors,  832- 
833  ; in  armature  circuits  for  speed 
regulation  of  induction  motors,  835- 
837  ; distributed,  932. 

Resonance,  condition  of,  234  ; electrical 
946. 

Reversing  polyphase  motors,  840-842. 

Revolving  armatures,  77,  92. 

Rheostats,  starting  and  regulating,  for 
induction  motors,  832-833. 

Ring  armature,  92. 

Ring  winding,  78. 

Roessler,  on  dependence  of  iron  losses 
of  transformer  upon  form  factor, 
506. 

Roiti,  cited,  594. 

Rotary  converters,  83,  755  ; heating  of 
armature  conductors  in  765-772  ; 
connecting  up,  777-778. 

Rotary  field  induction  motors,  784- 
785 ; currents,  torque,  impedance, 
and  magnetic  leakage  in,  800-803 ; 
vector  relations  in  the,  803-807 ; 
substituted  impedance  for  the,  807- 
815 ; as  an  asynchronous  generator, 
822-824 ; features  of  construction, 
824  ff. 

Rotating-field  alternators,  77,  125-126. 

Rotating  magnetic  field,  619,  785-790; 
action  of  a short-circuited  armature 
winding  within  a,  790-792. 

Rotating  vector,  257-258. 

Rotor,  defined,  791-792. 

Rushes,  current,  583-5S4. 

Russell,  Alternating  Currents,  cited,  430, 
904,  931. 

Ryan,  experiment  showing  effect  of 
magnetic  leakage,  466  ; tracing  trans- 
former curves,  594. 

Ryan  and  Bedell,  synchronous  motor 
experiment,  714. 

Ryan  and  Merritt,  testing  transformers, 
650. 


INDEX 


965 


s 

Saturation  curve,  621-624. 

Scalar  value  of  two  forces,  8. 

Screening  action  due  to  eddy  currents, 
427-433. 

Searing,  tracing  curves,  649. 

Secondary  induced  voltage  of  induction 
motor,  798-800. 

Secondary  winding,  effect  of  current  in, 
on  forms  of  primary  current  waves, 
481-483. 

Secondary  windings,  791. 

Self-excited  alternator,  107, 

Self-inductance,  134-136;  the  henry  the 
unit  for  measuring,  136 ; of  a short 
coil,  136-137  ; of  a circuit  containing 
variable  permeability,  137-138 ; ex- 
amples of  values  of,  141-146 ; rate 
of  expenditure  of  work  in  circuit 
containing,  183-185  ; vector  relations 
of  current  and  voltage  in  circuit  con- 
taining resistance,  capacity,  and,  in 
series,  209-211 ; of  alternators,  609— 
611  ; of  parallel  wires,  897-907 ; 
relations  of  distributed  resistance, 
leakage  conductance,  electrostatic 
capacity,  and,  932-945. 

Self-induction,  62,  128-130 ; coefficient 
of  (self -inductance),  135. 

Self-inductive  circuit,  vector  diagrams 
of  voltage  relations  in,  130-132 ; 
current  in,  146-149 ; effect  of  eddy 
currents  and  of  hysteresis  on  rise  and 
fall  of  current  in,  165—166 ; high 
voltage  generated  on  breaking  a, 
166-167. 

Separately  excited  alternator,  107. 

Series,  mutual  inductance  in  coils  in, 
456-459 ; alternators  in,  664-668. 

Series  alternating  current  motors,  874- 
881. 

Series  circuits,  261 ; graphical  and 
analytical  treatment  of  problems 
relating  to,  262-275;  conclusions  in 
regard  to,  275-277  ; solution  of,  com- 
bined with  parallel  circuits,  302-309. 

Series  motors,  784 ; vector  diagrams  of, 
and  expressions  for  voltage,  881-887. 

Series  repulsion  motor,  878-879. 

Series  transformers,  75-76,  571-575. 

Series-wound  alternator,  107. 

Shading  coils,  853. 

Shape  of  voltage  and  current  waves 
used  in  testing  transformers,  593-594. 

Sheet-iron  plates,  eddy  current  loss  in, 
421^24. 

Shell  type  transformers,  532,  534-535. 

Short-circuit  current  curve,  632. 

Short-circuited  coils,  independent,  in 
windings  of  rotating  field  induction 
motors,  824. 


Short-circuiting  of  terminals  of  second- 
ary windings,  49.5 — 198. 

Shunt,  use  of.  in  amperemeters,  72-73, 
76. 

Shunt-wound  alternator,  107. 

Siemens  electro-dynamometer,  66—67. 

Single-phase  induction  motors,  843- 
850 ; locus  diagram  of,  and  sub- 
stituted impedance,  850—852  ; start- 
ing, 852-853. 

Single-phase  machines,  55. 

Sinusoidal  voltage  in  a self-inductive 
circuit,  133-134. 

Skin  effect,  903,  910. 

Slip  of  induction  motor,  798,  815,  S56. 

Slip  rings,  19. 

Smithsonian  Mathematical  Tables,  93S. 

Solenoid,  self-inductance  of  a,  136—137. 

Sparking,  avoidance  of,  at  rectifying 
commutator,  115—117. 

Specific  inductive  capacity.  169. 

Speeds  of  induction  motor,  797-798. 

Split  dynamometer  methods  of  measur- 
ing power,  357-358. 

Split-pole  converters,  773. 

Squirrel  cage  winding,  791 ; of  induc- 
tion motors,  824. 

Standardization  Rules  of  American 
Institute  of  Electrical  Engineers, 
585—586,  742. 

Standing  torque,  856. 

Stanley  alternator,  118-119. 

Star-connected  systems,  361-363. 

Starting  devices,  polyphase  induction 
motors,  832  ff. 

Starting  single-phase  induction  motors, 
852-853. 

Starting  torque,  856. 

Star  winding,  58. 

Stator,  defined,  792. 

Steel,  constants  of  magnetic  hysteresis 
related  to,  409 ; for  use  in  trans- 
formers, 509-515. 

Steinmetz,  cited,  13,  423,  506,  906; 
experiments  by,  122,  407 ; experi- 
ments showing  energy  loss  due  to 
hysteresis,  403 ; hysteretie  angle  of 
advance  of,  435.  442  : parallel  opera- 
tion of  alternators,  689. 

Step,  voltage  waves  in.  664. 

Step  by  step  method  of  testing  for 
hysteresis  loss,  416. 

Stray  power  methods,  for  obtaining 
transformer  efficiency,  587-590 ; of 
testing  alternators,  729—732 : of 

testing  induction  motors,  855-856. 

Substituted  impedance,  for  the  rotary 
field  induction  motor,  807-815  : locus 
diagram  of  single-phase  motor  and, 
850-852. 

Sumpner,  354,  356. 

Surges,  current,  583-584. 


966 


INDEX 


Susceptance,  of  a circuit,  214 ; of  an 
admittance,  253. 

Swenson  and  Frankenfield,  table  . of 
hysteresis  constants  compiled  by,  409. 

Swinburne,  Hedgehog  transformer  of, 
517. 

Switchboards  for  connecting  alternators 
in  parallel,  699-702. 

Synchronism,  voltage  waves  in,  664. 

Synchronizers  and  synchronizing,  690- 
699. 

Synchronizing  current  of  alternators, 
669-676. 

Synchronizing  lamps,  691-694. 

Synchronous  condenser,  748-754. 

Synchronous  generators,  19. 

Synchronous  impedance,  621,  632-634, 
*706-715. 

Synchronous  machines,  597  ff. 

Synchronous  motors,  alternators  as, 
702-704 ; relation  of  voltages,  cur- 
rents, and  power  in,  704-715 ; locus 
diagrams  of  currents,  voltages,  and 
loads  of,  715-724 ; application  of 
diagrams  to  machines  wound  with 
any  number  of  phases,  728-729  ; for 
testing  alternators,  738-739 ; hunt- 
ing of,  743-748  ; asynchronous  motors 
contrasted  with,  784. 

Synchronous  regulator,  774. 

Synchroscopes,  696-699. 

T 

Table,  characteristic  features  of  dif- 
ferent forms  of  alternating-current 
and  voltage  curves,  43 ; effect  of 
distributed  coils  on  armature  voltage, 
52 ; specific  resistance  of  insulators, 
122  ; formulas  for  dielectric  materials, 
122 ; polarized  relays,  143 ; self-in- 
ductance of  armature  in  place,  145 ; 
power  factors  and  reactive  factors, 
342 ; hysteresis  constants,  409 ; 
Ewing’s,  of  ratio  of  magnetic  force 
at  various  depths  in  a plate  to 
magnetic  force  at  surface,  431 ; 
of  iron  and  copper  losses  in  trans- 
formers, 520  ; of  dielectric  strength 
of  alternators,  741 ; relation  of 
number  of  poles  to  synchronous 
speed  in  induction  motors,  826 ; 
self-inductances  of  solid  cylindrical 
copper  conductors,  901 ; distribution 
of  current,  910. 

Tandem  connection,  induction  motors, 
838-840. 

Teaser  winding,  554,  561. 

Tee  connection  of  transformers,  554- 
555. 

Teeth,  armature,  94,  124 ; of  rotating 
field  induction  motors,  831. 


Telephone  service,  long  distance,  943- 
945. 

Tensor  of  vector  quantity,  242. 

Terminals  of  transformer,  insulation 
of,  545. 

Tesla,  rotary  field  motor  applications, 
784. 

Testing,  transformers,  585-586;  alter- 
nators, 729  ff. ; induction  motors, 
853-859. 

Thompson,  S.  P.,  80,  81. 

Thomson,  Elihu,  538,  567 ; on  electro- 
magnetic repulsion,  864. 

Thomson,  J.  J.,  formulas  showing 
extent  of  magnetic  screening  in 
stampings,  430 ; current  density  at 
any  point  within  a conductor, 
910. 

Thomson  alternating-current  ampere- 
meter, 74. 

Thomson-Houston  alternator,  93. 

Thomson  impedance  coils,  575-576. 

Three-instrument  methods  of  measuring 
power,  354-358. 

Three-phase  machine,  56-57. 

Three-phase  systems,  methods  of  con- 
nection, 361—367 ; measurement  of 
power  in,  389,  391. 

Time  constants  of  a circuit,  206-208 ; 
examples  of,  208-209. 

Tobey  and  Walbridge,  article  by,  cited, 
638. 

Todhunter,  cited,  368. 

Torque,  in  rotary  field  induction  motor, 
801 ; of  polyphase  induction  motor, 
815  ff. ; standing  and  starting,  of 
induction  motors,  856. 

Torsion  head,  66-67. 

Transference  of  electricity,  in  self-in- 
ductive circuits,  149-152  ; transient, 
in  divided  circuits,  162-164;  by  the 
effect  of  mutual  induction,  456- 
463. 

Transformation,  ratio  of,  in  converters, 
758-763. 

Transformer  oil,  539. 

Transformers,  fundamental  principle 
of,  61-64 ; definition,  64 ; losses 
in  operation  of,  64-65 ; mutual 
induction  by  means  of,  443  ; diagrams 
of,  443-449  ; ratio  of  transformation 
in,  463—465 ; magnetic  leakage  in, 
465—168 ; constant  current,  46S ; 
constant  voltage,  468 ; diagrams  of, 
with  magnetic  leakage,  468—475; 
exciting  current,  475-477 ; effects 
of  variable  reluctance,  of  hysteresis, 
and  of  eddy  currents  on  form  of 
primary  current  wave,  477-481  ; 
forms  of  primary  current  waves  as 
affected  by  current  in  secondary 
winding,  481-483 ; core  magnetiza- 


INDEX 


967 


tion,  483-486 ; circle  diagram  for 
non-reactive  secondary  circuit,  and 
active  voltage  locus,  486 ; circle  dia-  j 
gram  where  transformer  load  con- 
tains constant  reactance  and  variable 
resistance,  490 ; locus  of  current 
when  load  reactance  is  varied  and 
resistance  is  kept  constant,  498-501  ; 
use  of  equivalent  impedances  in  solv- 
ing problems,  501-505 ; effect  of 
harmonics  in  waves  of  voltage  and 
current  upon  operation,  505-507 ; 
effects  of  changes  of  frequency  and 
voltage,  507-509  ; iron  and  steel  for, 
509-515;  alloying  materials,  515; 
efficiencies  of,  515-523 ; Hedgehog, 
517  ; U.  S.  table  of  iron  and  copper 
losses,  520 ; regulation  of,  522- 
523;  polyphase,  524-531;  construc- 
tive features  of  constant  voltage 
transformers,  531  ff.  ; core  type  and 
shell  type,  532-537 ; cooling,  538- 
544;  insolation  in,  544-546;  connect- 
ing constant  voltage,  and  features 
of  their  operation,  546  ff. ; polarity 
of  windings,  547 ; open  delta  on  V 
connection,  552  ; tee  connection,  554  ; 
teaser  windings,  554,  561  ; phase 
transformation,  561-566;  double  delta 
connection,  563;  transformation  from 
constant  voltage  to  constant  current, 
566  ; series  or  current  transformers, 
571-575;  reactance  coils,  impedance 
coils,  or  choking  coils,  575-576; 
autotransformers  or  compensators, 
576-581  ; calculation  of  magnetic 
leakage,  581-583 ; current  rushes 
and  surges,  583-584 ; methods  of 
testing,  585  ff.  ; wattmeter  method, 
586-587 ; stray  power  methods, 
587  ; regulation  tests,  590-591  ; heat- 
ing tests,  591-592  ; dielectric  strength 
of  insulation,  592-593 ; determina- 
tion of  wave  shape  in  testing,  593- 
594 ; methods  used  in  historically 
important  tests  of,  594-596. 

Transient  state  in  a circuit,  effect  of 
self-inductance  and  capacity  on, 
186-187. 

Tri-phase  transformers,  524-527. 

True  power,  325. 

Tuned  circuits,  947. 

Turbine  generators,  rotating  field  for, 
125-127. 

Two-circuit  windings,  80. 

Two-phase  machine,  55-56 ; method 
of  converting  a direct-current  ma- 
chine into,  83. 

Two-phase  system,  voltage  curves  of, 
360 ; methods  of  connection,  360- 
361  ; measurement  of  power  in,  389- 
391. 


U 

Undercompensated  series  motor,  878. 

1 Underground  cables,  transmission  of 
power  over,  942-945. 

Uniformly  rotating  magnetic  field,  786- 
788. 

V 

Variation  of  angular  velocity,  effect  of, 
on  parallel  operation  of  alternators 
and  division  of  load,  684-690. 

V connection  of  transformers,  552-554. 

V-curves,  686. 

Vector,  envelope  of  the  current,  when 
conditions  in  circuit  vary,  237-240. 

Vector  analysis  and  the  complex 
quantity,  241-244. 

Vector  diagram,  260. 

Vector  diagrams,  representing  voltage 
relations  in  an  inductive  alternating- 
current  circuit,  130-132 ; showing 
voltage  and  current  relations  in  a 
charged  condenser,  176-180;  of  alter- 
nating-current series  motors,  881- 
887. 

Vector  formulas  in  solving  transformer 
problems,  502-504. 

Vector  polygon,  12,  260.  . 

Vector  power,  333. 

Vector  quantities,  8. 

Vector  relations,  of  current  and  voltage 
in  circuit  containing  resistance,  self- 
inductance, and  capacity  in  series, 
209-211;  in  rotating  field  induction 
motor,  803-807. 

Vectors,  8 ; rotating,  9 ; relations 
of  their  components  and,  15-17 ; 
addition  and  subtraction  of,  244 ; 
multiplication  and  division  of,  244- 
246 ; complementary,  246  ; recipro- 
cals of,  248 ; involution  and  evolu- 
tion of,  249-251  ; differentiation  and 
integration  of,  251-252 ; graphical 
combination  of,  257 ; expression  of 
power  relations  by  means  of,  333- 
339. 

Vector  triangle,  12. 

Ventilation,  armature,  124-126 ; of 
armature  alternators,  600. 

Versor  of  an  expression,  242. 

Vibrator,  oscillograph,  647. 

Vinculum,  use  of,  255. 

Voltage,  alternating,  2 ; resolution  of 
irregular  waves  of,  into  their  har- 
monics, 33-35 ; instruments  for 
measuring  alternating,  43-45 ; com- 
parison of  that  developed  by  an 
alternator  and  that  developed  by  a 
direct-current  dynamo,  46 ; effect 
of  arrangements  of  windings  on  that 
of  armatures,  47-54  ; of  self-indue- 


968 


INDEX 


tion,  128 ; impressed,  129 ; genera- 
tion of  high,  by  breaking  a self- 
inductive  circuit,  166-167  ; capacity 
or  condenser,  177,  267 ; effects  of 
changes  of,  on  transformers,  507- 
509. 

Voltages,  vector  diagrams  representing, 
130-132 ; vector  relations  of,  in 
circuit  containing  resistance  and 
inductance,  149 ; relations  between 
currents  and,  in  polyphase  systems, 
370. 

Voltage  coil,  345. 

Voltage  control  of  converters,  773. 

Voltage  curve  of  alternator,  22-27,  621, 
634  ff. ; determining  the  effective 
value  of  an  irregular,  42-43 ; methods 
for  determining  form  of,  644-652 ; 
effect  of  form  of,  on  parallel  operation 
and  division  of  loads,  689-690. 

Voltage  limitations  in  converters,  763- 
765. 

Voltage  regulators,  575-581,  654-661. 

Voltage  transformer,  75-76. 

Voltage  waves,  irregular  current  and, 
expressed  as  complex  quantities, 
226-229. 

Volt-amperes,  325 ; reactive,  334. 

Voltmeters,  alternating-current,  43-45, 
65-75;  ar\  ,'angement  of,  for  measur- 
ing high  voltages  or  large  currents,  76. 

W 

Warburg,  on  energy  losses  due  to 
hysteresis,  403-404. 

Warren  alternator,  119. 

Water-cooled  transformers,  542-543. 

Wattless  current,  314. 

Wattmeter,  alternating-current,  65-75, 
313 ; arrangement  for  measuring 
high  voltages  or  large  currents,  75-76  ; 
for  measuring  power  in  alternating 
circuit,  344-351 ; errors  in  readings, 
due  to  reactance  of  coils,  346-349 ; 
correction  of  readings,  on  account  of 
power  absorbed  by  instrument,  349- 
350  ; effect  of  eddy  currents  in  frame, 
350-351 ; electrostatic,  353-354  ; for 
measuring  power  in  polyphase  sys- 
tems, 389  ff. 


Wattmeter  method  of  testing  trans- 
formers, 586-587. 

Watts  expended  in  a circuit,  325. 

Wave  form  of  voltage  produced  by  an 
alternator,  742. 

Wave  length  constant,  939. 

Waves,  examples  of  irregular,  28-32. 

Wave  shape,  determination  of,  in  test- 
ing transformers,  593-594. 

Wave  windings,  81-82. 

Wavy  field  current  in  alternator,  116, 
117. 

Westinghouse  alternator,  93,  126. 

Westinghouse  rotating  field  magnet, 
126. 

Weston  alternating-current  voltmeter, 
67-69. 

Weston  wattmeter,  350. 

Wilde,  operation  of  alternators  in 
parallel,  669. 

Wilkes,  tracing  curves,  649. 

Williamson,  Differential  Calculus,  cited, 
242. 

Windage,  losses  in  alternators  due  to, 
597. 

Windings,  armature,  47  ff. ; single  and 
polyphase,  55 ; mesh  or  delta,  57 ; 
star  or  Y,  58 ; methods  of  connecting 
up,  58-60 ; drum  and  ring,  78 ; 
bar  and  coil,  78 ; distributed,  78, 
85-90 ; lap,  80 ; two-circuit,  80 ; 
wave  or  progressive,  81-82 ; one- 
circuit,  82 ; barrel,  86 ; polarity  of, 
547-548 ; teaser,  554,  561 ; amor- 
tisseur,  747 ; armature,  in  rotating 
field,  790-792  ; primary  and  second- 
ary, 791 ; of  induction  motors, 
824-832. 

Wood,  H.  P.,  cited,  373. 

Working  resistance  and  reactance,  221. 

Y 

Y-box  for  measuring  power,  395. 

Y-connection,  361,  363;  for  more  than 
three  phases,  36S-370. 

Y-winding,  58. 

Z 

Zipernowsky  alternator,  arrangement  of 
commutator  in,  115-116. 


Printed  in  the  United  States  of  America, 


' 


Date  Due 


i .....  _ 

MAR  2 41< 

48 

g|p1  9'S 

3 

L.  B.  Cat.  No.  1137 


u 


54537 


f 


